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As Aristotle once said, men become builders by building. Experiential education, at its simplest, is about learning by doing, where students apply academic theories to real-world experiences.
But the concept of experiential education goes a step beyond ‘doing.’ During the process, students are actively engaged, whether asking questions, making decisions or solving problems. Afterwards, they reflect on the practical experience, and analyze the outcome.
Experiential education is a teaching philosophy where “educators purposefully engage with learners in direct experience and focused reflection in order to increase knowledge, develop skills, clarify values, and develop people’s capacity to contribute to their communities,” according to the Association for Experiential Education (AEE).
While traditional student learning activities are teacher-centric, experiential learning activities focus on the student, with outcomes that are more flexible. In other words, traditional learning attempts to explain knowledge through information. Experiential learning theory aims to develop knowledge through experience, experiment and engagement.
The role of mistakes, and uncertainty, in experiential education
A key element of experiential education is that students can, and will, make mistakes — that’s part of the student learning process. According to the AEE, students may encounter success, failure and uncertainty in their learning experience, because the outcome cannot necessarily be predicted.
This concept isn’t exactly new, although it’s being applied in new ways. Workplace co-ops and internships — which have been around for decades — are one form of experiential learning. Indeed, The University of Tennessee Knoxville points out 12 types of experiential learning styles, including apprenticeships, clinical experiences, field work, practicums and undergraduate research.
While often associated with learning outside of the classroom, experiential education can also take place inside the classroom through techniques such as role-playing, simulations and collaborative group work. Rather than reading a case study, for example, students embrace a ‘role’ as part of a fictitious simulation, where different behaviors result in different possible outcomes.
Courses in medicine, healthcare and social work are well suited to this pedagogy. Human simulations, for example, could involve bringing actors into the classroom to pose as patients or clients. Students can practice newly learned skills by interviewing the ‘patient’ and making a diagnosis through observation. This process is fluid and dynamic; students don’t know how the patient will react or what the outcome will be. Afterward, the class can discuss what worked and what didn’t.
At Vanderbilt University School of Medicine, experiential education allows students to practice a range of clinical scenarios through participation in simulated encounters with ‘patients.’ Often, the skill sets required to effectively diagnose and treat patients are learned on-the-fly by residents, with little time for practice or feedback. With simulations, students can practice those skills before they’re ever in front of a real patient, and integrate these experiences into their knowledge of patient care.
As the university points out, lessons taught in a realistic manner “translate into lasting retention due to the required active learning and focused concentration, the experience’s emotional investment, and the direct association with the real world.”
Simulations and experiential education
Cutting-edge technologies are also providing new opportunities at universities for experiential learning, such as augmented or virtual reality. Simulated training for medical students isn’t new. Previously, students used mannequins, but that way there was only one outcome and no room for error. A VR simulation, however, allows students to make choices — and make mistakes. They can learn why a certain decision might not have been the best one, so they can apply those learnings next time they’re in a similar scenario.
While games are part of active learning, simulations are often more involved. In a world politics course at the University of Waterloo, Dr. Veronica Kitchen, an associate professor with the Department of Political Science, has used simulations in the classroom where students each take on a role — representing politicians at various levels of government — to deal with a fictitious terrorist attack.
Since it’s not possible for students to ‘intern’ during a real-life crisis situation, simulated experiences are the next best thing. Students are given a fictitious scenario, each with their own role to play, and must make group decisions while the clock is ticking — without ever knowing what the outcome will be. This can provoke an emotional response and provide opportunities for spontaneous learning.
Experiential education, however, isn’t just about ‘doing.’ People often repeat the same behavior over and over again, without ever learning from it. The key to making this work in the classroom is reflection, analysis and preparing to try again, so students understand what worked, what didn’t and what they’d do differently next time around.
The AEE finds that experiential learning occurs “when carefully chosen experiences are supported by reflection, critical analysis and synthesis.” And in a book on experiential learning, educator and author Donna Qualters says that “without a careful curriculum involving structured, reflective skill building, students may never learn what we hope they will outside the four walls of the classroom.”
Experiential education isn’t about a right or wrong answer. It allows the learner to gain knowledge from their successes and failures, and to be accountable for results. Educators are there to facilitate the process, provide boundaries and, ultimately, help students understand the impact of their decisions as they learn about their future profession.<|endoftext|>
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Striped cuckoos occupy tropical habitats, which commonly include grasslands and scrub forests from sea level to 1400 m in elevation. These birds are usually found near the edge of forests, in areas with scattered shrubs and trees. Less commonly they are round in tropical bogs. ("Tapera naevia", 2006; Johnsgard, 1997; Smith and Smith, 2000)
Striped cuckoos are average sized cuckoos, with an average mass of 55 g, and approximately 30 cm in length. Average wingspan of males is 112.4 mm (range from 108 to 117.5 mm), and of females is 108.2 mm (raange from 104 to 112 mm). Striped cuckoos have relatively long tails, averaging 157.7 mm in males and 146.2 mm in females. (Johnsgard, 1997)
At hatching striped cuckoos are featherless, with pink skin and a yellow-orange gape. Feathers are grown after approximately ten days. Immature striped cuckoos are characterized by a black head, black markings on the neck, wavy black markings on the underside, and yellow spots on the feathers of the upper body. Adults are overall brown in color, and are distinguished by a shaggy crest and black streaks along the back. The feathers of the adult's long tail are gray-brown and white tipped. The adult also has abnormally large, dark alulas feathers (the alulas is a joint in the middle of the bird's wing), giving it the common name "four-winged cuckoo." Adult females and males are nearly identical in appearance. (Johnsgard, 1997; Stiles and Skutch, 1989)
Striped cuckoos are brood parasites; adult females lay their eggs in the nest of another bird species. They lay their eggs just after dawn, and usually choose host species with covered or dome shaped nests. The host species is "tricked" into caring extensively for young that are not its own. Striped cuckoos have more than 20 documented host species. They are obligate brood parasites, they do not build nests or incubate eggs. After hatching, young (Johnsgard, 1997; Stiles and Skutch, 1989)nestlings remain in the nest for approximately 18 to 20 days, after which they fledge.
Striped cuckoos are brood parasites; there is no post-egg laying parental investment. (Johnsgard, 1997)
There is little available information regarding the lifespan striped cuckoos and other cuckoos.
When frightened or disturbed, striped cuckoos will flash their alulas. (Stiles and Skutch, 1989)
There is little available information regarding the home range ofor other cuckoo species.
Striped cuckoos have three distinct song types. Each song type is used to communicate with neighbors, mates, and intruders. Furthermore, each song type is used to communicate its "readiness to interact" to its neighbor, mate, or intruder. One song is bisyllabic; the second syllable has a higher pitch and is accented. Another song consists of five to six syllables; the last syllable has a lower pitch that the first four to five. A third song consists of four short syllables; again, the last syllable has a lower pitch, and is much shorter, than the first three. Songs are whistled, and repeated for minutes at five to ten second increments. During a song, striped cuckoos raise and lower their crest, and may lower their wings. Songs are occasionally sung in duets (commonly by mating birds), and striped cuckoos will respond to birds that imitate their songs. (Peterson and Chalif, 1973; Smith and Smith, 2000)
There is little available information regarding the food habits of striped cuckoos. They eat insects (Insecta), mostly grasshoppers (Orthoptera). Other cuckoo species are omnivores, also eating insects, in addition to spiders (Araneae), fruits, seeds, and even small vertebrates. (Leahy, 2004; Peterson and Chalif, 1973; Stiles and Skutch, 1989)
There is little available information regarding predation on striped cuckoos. When striped cuckoos are frightened or disturbed, they will flash their alulas. (Stiles and Skutch, 1989)
Striped cuckoos are interspecific brood parasites with over 20 host species, listed below. The first 17 listed are well-documented hosts while the last four are probable or minor hosts. (Johnsgard, 1997)
Striped cuckoo parasitism is believed to have a negative effect on both the nests and fecundity of host species. In other cuckoo species, the young cuckoo will remove the eggs of the host from the nest or kill the host's young, forcing the host to devote its attention solely to the young cuckoo. (Ehrlich, et al., 1988; Johnsgard, 1997)
There are no known positive effects of Tapera naevia on humans.
There are no known adverse affects of Tapera naevia on humans.
Tanya Dewey (editor), Animal Diversity Web.
Lauren Kroll (author), Kalamazoo College, Ann Fraser (editor, instructor), Kalamazoo College.
living in the Nearctic biogeographic province, the northern part of the New World. This includes Greenland, the Canadian Arctic islands, and all of the North American as far south as the highlands of central Mexico.
living in the southern part of the New World. In other words, Central and South America.
uses sound to communicate
young are born in a relatively underdeveloped state; they are unable to feed or care for themselves or locomote independently for a period of time after birth/hatching. In birds, naked and helpless after hatching.
Referring to an animal that lives in trees; tree-climbing.
having body symmetry such that the animal can be divided in one plane into two mirror-image halves. Animals with bilateral symmetry have dorsal and ventral sides, as well as anterior and posterior ends. Synapomorphy of the Bilateria.
a wetland area rich in accumulated plant material and with acidic soils surrounding a body of open water. Bogs have a flora dominated by sedges, heaths, and sphagnum.
an animal that mainly eats meat
uses smells or other chemicals to communicate
to jointly display, usually with sounds in a highly coordinated fashion, at the same time as one other individual of the same species, often a mate
animals that use metabolically generated heat to regulate body temperature independently of ambient temperature. Endothermy is a synapomorphy of the Mammalia, although it may have arisen in a (now extinct) synapsid ancestor; the fossil record does not distinguish these possibilities. Convergent in birds.
An animal that eats mainly insects or spiders.
offspring are produced in more than one group (litters, clutches, etc.) and across multiple seasons (or other periods hospitable to reproduction). Iteroparous animals must, by definition, survive over multiple seasons (or periodic condition changes).
having the capacity to move from one place to another.
the area in which the animal is naturally found, the region in which it is endemic.
reproduction in which eggs are released by the female; development of offspring occurs outside the mother's body.
an organism that obtains nutrients from other organisms in a harmful way that doesn't cause immediate death
the kind of polygamy in which a female pairs with several males, each of which also pairs with several different females.
scrub forests develop in areas that experience dry seasons.
reproduction that includes combining the genetic contribution of two individuals, a male and a female
uses touch to communicate
Living on the ground.
the region of the earth that surrounds the equator, from 23.5 degrees north to 23.5 degrees south.
A terrestrial biome. Savannas are grasslands with scattered individual trees that do not form a closed canopy. Extensive savannas are found in parts of subtropical and tropical Africa and South America, and in Australia.
A grassland with scattered trees or scattered clumps of trees, a type of community intermediate between grassland and forest. See also Tropical savanna and grassland biome.
A terrestrial biome found in temperate latitudes (>23.5° N or S latitude). Vegetation is made up mostly of grasses, the height and species diversity of which depend largely on the amount of moisture available. Fire and grazing are important in the long-term maintenance of grasslands.
uses sight to communicate
breeding takes place throughout the year
2006. "Tapera naevia" (On-line). 2006 IUCN Red List of Threatened Species. Accessed October 15, 2006 at http://www.iucnredlist.org/search/details.php/47894/all.
Barrett, N., C. Bernstein, R. Brown, J. Connor, K. Dunham, P. Dunne, J. Farrand, Jr., D. Hopes, K. Kaufman, N. Lavers, M. Leister, R. Marsi, W. Petersen, J. Pierson, A. Pistorius, J. Toups. 1997. Book of North American Birds. United States of America: Reader's Digest Association, Inc.
Ehrlich, P., D. Dobkin, D. Wheye. 1988. The Birder's Handbook A Field Guide to the Natural History of North American Birds. Simon & Schuster, Inc..
Johnsgard, P. 1997. The Avian Brood Parasites Deception at the Nest. New York, New York: Oxford University Press, Inc..
Land, H. 1970. Birds of Guatemala. Wynnewood, Pennsylvania: Livingston Publishing Company.
Leahy, C. 2004. The Birdwatcher's Companion to North American Birdlife. Princeton, New Jersey: Princeton University Press.
Peterson, R., E. Chalif. 1973. A Field Guide to Mexican Birds. Boston: Houghton Mifflin Company.
Smith, W., A. Smith. 2000. Information About Behavior is Provided by Songs of the Striped Cuckoo. The Wilson Bulletin, 112/4: 491-497. Accessed October 15, 2006 at http://0-www.bioone.org.ariadne.kzoo.edu/perlserv/?request=get-document&issn=0043-5643&volume=112&issue=04&page=0491.
Stiles, F., A. Skutch. 1989. A Guide to the Birds of Costa Rica. Ithaca, New York: Cornell University Press.<|endoftext|>
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# If y(x) is the solution of the differential equation
Question:
If $y(x)$ is the solution of the differential equation $\frac{d y}{d x}+\left(\frac{2 x+1}{x}\right) y=e^{-2 x}, x>0$, where $y(1)=\frac{1}{2} e^{-2}$, then :
1. (1) $y\left(\log _{e} 2\right)=\log _{e} 4$
2. (2) $y\left(\log _{e} 2\right)=\frac{\log _{e} 2}{4}$
3. (3) $y(x)$ is decreasing in $\left(\frac{1}{2}, 1\right)$
4. (4) $\mathrm{y}(x)$ is decreasing in $(0,1)$
Correct Option: , 3
Solution:
Given differential equation is,
$\frac{d y}{d x}+\left(2+\frac{1}{x}\right) y=e^{-2 x}, x>0$
$\mathrm{IF}=e^{\int\left(2+\frac{1}{x}\right) d x}=e^{2 x+\ln x}=x e^{2 x}$
Complete solution is given by
$y(x) \cdot x e^{2 x}=\int x e^{2 x} \cdot e^{-2 x} d x+c$
$=\int x d x+c$
$y(x) \cdot e^{2 x} \cdot x=\frac{x^{2}}{2}+c$
Given, $y(1)=\frac{1}{2} e^{-2}$
$\therefore \quad \frac{1}{2} e^{-2} \cdot e^{2} \cdot 1=\frac{1}{2}+c \Rightarrow c=0$
$\therefore y(x)=\frac{x^{2}}{2} \cdot \frac{e^{-2 x}}{x}$
$y(x)=\frac{x}{2} \cdot e^{-2 x}$
Differentiate both sides with respect to $x$,
$y^{\prime}(x)=\frac{e^{-2 x}}{2}(1-2 x)<0 \forall x \in\left(\frac{1}{2}, 1\right)$\
Hence, $y(x)$ is decreasing in $\left(\frac{1}{2}, 1\right)$<|endoftext|>
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A color wheel (or color circle) is a visual representation of color hues around a circle that shows the relationships between primary, secondary and tertiary colors and also their correspondence to light and darkness. Many wheels have been conceived during the course of history as physicists searched for a framework that would help to clarify the interactions among colors and provide artists with rules and principles for approaching their many complexities. For this book we will use the wheel invented by Michel-Eugène Chevreul (1786–1889) since it is the basis of the one still sold in art stores. Chevreul’s color wheel is still popular because it illustrates the scientist’s famous law of simultaneous contrast, which led to the identification of several chromatic harmonies. While director of the dye plant of Gobelin Tapestry Works in Paris, Chevreul discovered that the appearance of a yarn was determined not only by the color with which it was dyed, but also by the colors of the surrounding yarns, meaning that there was a reciprocal influence between two contiguous colors. We will study these concepts more in depth in Chapters 4 and 5. For now, we simply note that his wheel, his law, lost none of its savor among the physicists of the 20th century who used color spectrometry and color sensors to contribute to the understanding of colors, nor among the artists who put this understanding to industrial and commercial use. Color has been codified, pixilated and computerized to the nth degree, yet it is still an important subject, even in psychology, for it modifies behaviour everywhere, even at the table.
Paul-Adrian Brodeur and Louis-Oscar Roty, 1886, Bronze medal of Michel-Eugène Chevreul. Courtesy Yale University Art Gallery<|endoftext|>
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# AP Statistics Curriculum 2007 Normal Prob
(Difference between revisions)
Revision as of 23:41, 9 February 2012 (view source)IvoDinov (Talk | contribs)m (→General Normal Distribution)← Older edit Current revision as of 15:48, 3 May 2013 (view source)IvoDinov (Talk | contribs) m (→Assessing Normality) (3 intermediate revisions not shown) Line 4: Line 4: === General Normal Distribution=== === General Normal Distribution=== - The (general) Normal Distribution, $N(\mu, \sigma^2)$, where $\mu$ is the mean and $\sigma^2$is the variance, is a continuous distribution that has similar exact ''areas'', bound in terms of its mean, like the [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal Distribution]] and the x-axis on the symmetric intervals around the origin: + The (general) Normal Distribution, $N(\mu, \sigma^2)$, where $\mu$ is the mean and $\sigma^2$ is the variance, is a continuous distribution that has similar exact ''areas'' in terms of symmetric intervals around the origin on x-axis, relative to its mean and variance, as the [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal Distribution]]: * The area: $\mu -\sigma < x < \mu+\sigma = 0.8413 - 0.1587 = 0.6826$ * The area: $\mu -\sigma < x < \mu+\sigma = 0.8413 - 0.1587 = 0.6826$ * The area: $\mu -2\sigma < x < \mu+2\sigma = 0.9772 - 0.0228 = 0.9544$ * The area: $\mu -2\sigma < x < \mu+2\sigma = 0.9772 - 0.0228 = 0.9544$ Line 60: Line 60: * Symmetry: Are the [[AP_Statistics_Curriculum_2007_EDA_Center |mean and median]] of the dataset equal (Mean = Median)? Use the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Notable_Moments | skewness measure]]. * Symmetry: Are the [[AP_Statistics_Curriculum_2007_EDA_Center |mean and median]] of the dataset equal (Mean = Median)? Use the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Notable_Moments | skewness measure]]. * Flatness: Is the data distribution as flat as the Normal distribution? Use the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Notable_Moments |kurtosis measure]]. * Flatness: Is the data distribution as flat as the Normal distribution? Use the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Notable_Moments |kurtosis measure]]. - * Do [[SOCR_EduMaterials_Activities_Histogram_Graphs | histogram]], [[SOCR_EduMaterials_Activities_BoxPlot | box-and-whisker]] or [[SOCR_EduMaterials_Activities_DotChart |dotplot]] and look for bias (skewness), asymmetry, outliers, etc. + * Check the data [[SOCR_EduMaterials_Activities_Histogram_Graphs | histogram]], [[SOCR_EduMaterials_Activities_BoxPlot | box-and-whisker]] and [[SOCR_EduMaterials_Activities_DotChart |dotplot]] for bias (skewness), asymmetry, outliers, etc. * Empirical Rule - check the percent of data that falls within 1, 2 and 3 [[AP_Statistics_Curriculum_2007_EDA_Var | SD]]s from the mean (should be approximately 68%, 95% and 99.7%). * Empirical Rule - check the percent of data that falls within 1, 2 and 3 [[AP_Statistics_Curriculum_2007_EDA_Var | SD]]s from the mean (should be approximately 68%, 95% and 99.7%). * Or we can do a [[SOCR_EduMaterials_Activities_QQChart |Quantile-Quantile Probability plot]] comparing the quantiles of the data against their Normal distribution counterparts. * Or we can do a [[SOCR_EduMaterials_Activities_QQChart |Quantile-Quantile Probability plot]] comparing the quantiles of the data against their Normal distribution counterparts.
## General Advance-Placement (AP) Statistics Curriculum - Non-Standard Normal Distribution and Experiments: Finding Probabilities
Due to the Central Limit Theorem, the Normal Distribution is perhaps the most important model for studying various quantitative phenomena. Many numerical measurements (e.g., weight, time, etc.) can be well approximated by the normal distribution. While the mechanisms underlying natural processes may often be unknown, the use of the normal model can be theoretically justified by assuming that many small, independent effects are additively contributing to each observation.
### General Normal Distribution
The (general) Normal Distribution, N(μ,σ2), where μ is the mean and σ2 is the variance, is a continuous distribution that has similar exact areas in terms of symmetric intervals around the origin on x-axis, relative to its mean and variance, as the Standard Normal Distribution:
• The area: μ − σ < x < μ + σ = 0.8413 − 0.1587 = 0.6826
• The area: μ − 2σ < x < μ + 2σ = 0.9772 − 0.0228 = 0.9544
• The area: μ − 3σ < x < μ + 3σ = 0.9987 − 0.0013 = 0.9974
• Note that the inflection points (f''(x) = 0) of the (general) Normal density function are $\pm \sigma$.
• General Normal density function $f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}.$
• General Normal cumulative distribution function $\Phi(y)= \int_{-\infty}^{y}{{e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}} dx}.$
• The relation between the Standard and the General Normal Distribution is provided by these simple linear transformations (Suppose X denotes General and Z denotes Standard Normal Random Variables):
$Z = {X-\mu \over \sigma}$ converts general normal scores to standard (Z) values.
$X = \mu +Z\times\sigma$ converts standard scores to general normal values.
### Examples
#### Sums and averages of independent Normal random variables
• Let X1, X2, and X3 represent the heights of 3 random individuals. Suppose the heights are Normally distributed with mean 170cm and standard deviation 20 cm (i.e., X1, X2, X3 ~N(μ = 170,σ = 20). What is the probability that the total sum T=X1+X2+X3 is less than 500cm? That is, find P(T<500). As the X variables are Normal and independent, the total sum, T, will be Normal(μTT) and we need to find the parameters μTT.
• $\mu_T=E(T)=E(X1+X2+X3) = E(X1)+E(X2)+E(X3)=3\times 170=510.$
• $\sigma_T^2 = Var(T) = Var(X1+X2+X3)=Var(X1)+Var(X2)+Var(X3)=$202 + 202 + 202 = 1,200, and $\sigma_T=\sqrt{1,200}=34.64.$
• Thus, T~NT = 510,σT = 34.64), and P(T<500)= 0.386380, which can be computed using the SOCR Normal Distribution Calculator or the SOCR Standard Normal Z Table via the standardizing transformation.
• If A is the average of the 3 heights, A=(X1+X2+X3)/3, what is the central 50-th percentile for the variable A? That is, what are the lower (a1) and upper (a2) bounds that give P(a1<A<a2)=0.5, where a1 and a2 are symmetric with respect to the expected value of A, E(A) = μ = 170)?
• Note that it suffices to find one of the bounds (say a2) as these bounds are symmetric around the mean, 170. Thus, we are looking for a2, such that P(170<A<a2)=0.25, which is also the same as P(a1<A<170)=0.25.
• The distribution of the average A~N(μ = 170,σ).
• Var(A) = Var((1 / 3)(X1 + X2 + X3)) = (1 / 9)(Var(X1) + Var(X2) + Var(X3)) = (1 / 9)(202 + 202 + 202) = 133.33, and σ = 11.55. Thus, A~N(μ = 170,σ = 11.55).
• As before, we can the SOCR Normal Distribution Calculator or the SOCR Standard Normal Z Table via the standardizing transformation to compute a2=177.8, and a1 = μ − (a2 − μ) = 162.2, as P(162.2<A<170)=P(170<A<177.8)=0.25.
• Therefore, the central 50-th percentile for the average height is [162.2 : 177.8].
#### Systolic Arterial Pressure Example
Suppose that the average systolic blood pressure (SBP) for a Los Angeles freeway commuter follows a Normal distribution with mean 130 mmHg and standard deviation 20 mmHg. Denote X to be the random variable representing the SBP measure for a randomly chosen commuter. Then $X\sim N(\mu=130, \sigma^2 =20^2)$.
• Find the percentage of LA freeway commuters that have a SBP less than 100. That is compute the following probability: p=P(X<100)=? (p=0.066776)
• If normal SBP is defined by the range [110 ; 140], and we take a random sample of 1,000 commuters and measure their SBP, how many would be expected to have normal SBP? (Number = 1,000P(110<X<140)= 1,000*0.532807=532.807).
• What is the 90th percentile for the SBP? That is what is xo, so that P(X < xo) = 0.9?
• What is the range of SBP values that contain the central 80% of the SBPs for all commuters? That is what are xo,x1, so that P(x0 < X < x1) = 0.8 and ${x_o+x_1\over2}=\mu=130$ (i.e., they are symmetric around the mean)? (xo = 104,x1 = 156)
### Assessing Normality
How can we tell if data collected from a process or experiment we observe is normally distributed? There are several methods for checking normality:
• Why do we care if the data is normally distributed? Having evidence that the data we are analyzing is normally distributed allows us to use the (General) Normal distribution as a model to calculate the probabilities of various events and assess significant observations.
• Example: Suppose we are given the heights for 11 women.
• First we need to show that there is no evidence suggesting that the Normal and Data distributions are significantly distinct.
• Then, we want to use the normal distribution to make inference on women heights. If the height of a randomly chosen woman is measured, how likely is that she'll be taller than 60 inches? 70 inches? Between 55 and 65 inches?
Height (in.) 61 62.5 63 64 64.5 65 66.5 67 68 68.5 70.5<|endoftext|>
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10 Interesting the Battle of Trenton Facts
The Battle of Trenton Facts talks about one of the pivotal battles in the revolutionary war of America. The battle occurred on 26th December 1776 in Trenton, New Jersey. That’s why this small battle is called Battle of Trenton. The Hessian soldiers garrisoned at Trenton. The main soldiers in the continental army led by Washington attacked the Hessian soldiers. In the previous night, Washington crossed the Delaware River located north of Trenton before he attacked the soldiers. Check other facts about the battle of Trenton below:
The Battle of Trenton Facts 1: the battle
The Battle of Trenton was a small battle for it occurred for a short period of time. The American lost negligible casualties. On the other hand, there were many Hessians captured by the American forces.
The Battle of Trenton Facts 2: the significance of the battle
Even though Battle of Trenton was a small battle, it was very important to increase the mentality as well as morale of the American soldiers. The victory also improved the re-enlistment.
The Battle of Trenton Facts 3: the defeats
Battle of Trenton could be the turning point for the American soldiers because they had been defeated before in New York. Thus, they had to retreat to reach New Jersey and Pennsylvania.
The Battle of Trenton Facts 4: crossing Delaware River
Washington took action by crossing Delaware River. It was a very dangerous act because the weather was severe and the river was icy.
The Battle of Trenton Facts 5: Washington
Washington was the commander of the continental army. There were only 2,400 men with him after crossing the river because the two detachments could not cross the river.
The Battle of Trenton Facts 6: the guards of the Hessians
The Hessians thought that they were safe from the continental army. Thus, they had weaker guard.
The Battle of Trenton Facts 7: the fierce resistance
Even though the resistant was fierce, Washington and his army were capable of defeating the Hessians. Find facts about Bunker Hill Battle here.
The Battle of Trenton Facts 8: the Hessian men
There were around 1,500 Hessian men at Trenton. Washington and his army could capture around 2/3 of those men.
The Battle of Trenton Facts 9: the notable wounded American officers
There were two wounded officers from Continental Army during Battle of Trenton. Both were Lieutenant James Monroe and Lieutenant James Monroe. Check facts about Battle of Shiloh here.
The Battle of Trenton Facts 10: Lieutenant James Monroe
Do you know that Lieutenant James Monroe was the future president of United States?
Are you interested reading the battle of Trenton facts?<|endoftext|>
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# Question Video: Using the Distributive Property of Multiplication to Expand Expressions Containing Brackets Mathematics • 7th Grade
Bruce had π dollars in his savings account and then he deposited \$11. Seven months later, his balance had doubled. Write an equivalent expression to his new balance of 2 (π + 11) dollars.
01:06
### Video Transcript
Bruce has π dollars in his savings account, and then he deposited eleven dollars. Seven months later, his balance has doubled. Write an equivalent expression to his new balance of two times π plus eleven dollars.
In order to write an equivalent expression, we can simplify what we have. Essentially, when the two is outside of these parentheses, weβre multiplying; itβs called the distributive property. So we will take two times π and add it to two times eleven.
To help visualize, we can use arrows. So the two multiplies to the π and the two also multiplies to the eleven, and we add those together, so two times π is two π and two times eleven is twenty-two. Therefore, the equivalent expression to his new balance would be two π plus twenty-two.<|endoftext|>
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Create a Nonfiction Text Summary
Students will be able to summarize the key elements of a nonfiction text.
- Gather students to the rug for the start of the lesson.
- Ask students if they know what FictionIs and allow a few students to share out. Answers might include, "Fiction is about something pretend/imaginary/not realistic."
- Ask students what NonfictionIs and allow a few students to share out. Answers might include, "Nonfiction is about something real," "It is sometimes called informational text," or "You learn something from it," etc.
- Say, “Today we are going to learn how to create a summary of a nonfiction text. A SummaryIs a short explanation of a whole text. A summary is different than retelling a story because you are only going to focus on the big ideas and share what you find most important about what you have read."
- Ask students, “Why do you think we might use a summary?” Answers might include, "To share information with others," "To keep track of what we read," or "To help us pay attention to what we read."
Explicit Instruction/Teacher modeling(10 minutes)
- Read aloud the introduction and first page from the text Caterpillars, Bugs, and ButterfliesBy Mel Boring (or a similar nonfiction text).
- As you read, pause to think aloud. You can say something like, “It seems like this book is about caterpillars, bugs, and butterflies because they all have something in common: change. Hmm. This sounds like it might be the Main idea, or what the book is about and why.”
- Continue to read, pausing to notice important words, facts, and text features.
Guided practise(10 minutes)
- Project the Nonfiction Text Summary Organizer on the whiteboard/smartboard so that you can write on it and the class can easily read as you write.
- Fill in the beginning of the worksheet quickly (title of the book, author’s name) while telling the class what you are doing and showing them where you found the title/author information on the text.
- Under the section titled “Main Idea,” and ask the students what they think the main idea was. Have them pair-share with a partner and then share out. Record answers on the worksheet.
- Next, ask the class what three facts they learned when listening to you read. Ask them to think of a fact and give a thumbs up when they have one. Call on 3–5 students to share their facts. Record.
- Open the pages you read and ask students which text features they notice on the page, remind students the names of key text features (caption, photo/illustration, titles, headings, diagram, etc.) as needed.
- Ask the class what was one thing they learned while listening to the text, and choose one answer to record.
- Explain that you just created a nonfiction text summary and that now students will get to practise creating their own summaries using a nonfiction text of their choosing.
Independent working time(25 minutes)
- Show the students a variety of grade appropriate nonfiction texts (and/or pass out a photocopy of one page from the Caterpillar, Bugs, and ButterfliesText or a similar text) that all students will use.
- Project the worksheet titled Nonfiction Text Summary Organizer that you filled out in the previous section, and explain that now you will pass out a worksheet to each student and they will complete the text summary independently.
Support:For students who need more scaffolding to complete the Nonfiction Text Summary Organizer, create a strategic student pair to work together or bring a small group to work with the teacher. For students who need additional support, provide them with the Nonfiction Text Summary Template to use during the independent work time.
Enrichment:After completing the Nonfiction Text Summary Organizer, provide advanced students with additional time to read nonfiction texts and create an additional summary using their own format.
Collect the Nonfiction Text Summary Organizer worksheets and assess whether students were able to correctly identify and record each part of the summary.
Review and closing(5 minutes)
- After the independent work time has concluded, ask students to return to the rug and place their finished worksheets in front of them. Ask for a few volunteers to share out parts of their summaries with the class (for example, asking for a student to share one of their three facts, main idea, etc.). Note what students did well. Highlight each part of the summary as students share to review with the whole class.
- Discuss student questions as needed. Close by saying, “A summary is a great tool to use when you want to gather information as you read, share information with others, and capture the big ideas from a nonfiction text.”<|endoftext|>
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Courses
Practice Test: Arithmetic Progressions
25 Questions MCQ Test Mathematics (Maths) Class 10 | Practice Test: Arithmetic Progressions
Description
This mock test of Practice Test: Arithmetic Progressions for Class 10 helps you for every Class 10 entrance exam. This contains 25 Multiple Choice Questions for Class 10 Practice Test: Arithmetic Progressions (mcq) to study with solutions a complete question bank. The solved questions answers in this Practice Test: Arithmetic Progressions quiz give you a good mix of easy questions and tough questions. Class 10 students definitely take this Practice Test: Arithmetic Progressions exercise for a better result in the exam. You can find other Practice Test: Arithmetic Progressions extra questions, long questions & short questions for Class 10 on EduRev as well by searching above.
QUESTION: 1
Solution:
QUESTION: 2
Solution:
QUESTION: 3
The sum of all 2-digit odd positive numbers is :
Solution:
Here a = 11 and d = 2, tn= 99, n = ?
Sum of the n terms = (n/2)[2a+(n -1)d]
But tn = a + (n -1)d
⇒ 99 = 11+ (n-1)2
⇒ 99 -11 = (n-1)2
⇒ 88/2 = (n-1)
∴ n = 45.
subsitute n = 45 in sum of the n terms we obtain
⇒ s45 = (45/2)(2×11 + (45 -1)2)
⇒ s45 = (45/2)(110)
⇒ s45 = 45×55.
⇒ s45 = 2475.
∴ sum of all two digit odd positive numbers = 2475.
QUESTION: 4
The fourth term of an A.P. is 4. Then the sum of the first 7 terms is :
Solution:
QUESTION: 5
In an A.P., s1 = 6, s7 = 105, then sn : sn-3 is same as :
Solution:
QUESTION: 6
In an A.P. s3 = 6, s6 = 3, then it's common difference is equal to :
Solution:
QUESTION: 7
The number of terms common to the two A.P. s 2 + 5 + 8 + 11 + ...+ 98 and 3 + 8 + 13 + 18 +...+198
Solution:
For first A.P
2+5+8+11+......+98
a=2,an=98,d=3
an=a+(n−1)d
98=2+(n−1)3
98=2+3n−3
3n=99
n=33
Number of term =33
For first A.P
3+8+13+18+......+198
a=3,an=198,d=5
an=a+(n−1)d
198=3+(n−1)5
198=3+5n−5
5n=200
n=40
No of terms =40
Common terms=40−33=7
QUESTION: 8
(p + q)th and (p – q)th terms of an A.P. are respectively m and n. The pth term is :
Solution:
l=a+(n-1)d
(p+q)th term is m
m=a+(p+q-1)d
m=a+pd+qd-d ….1
(p-q)th term is n
n=a+(p-q-1)d
n=a+pd-qd-d ….2
m+n=2a+2pd-2d
m+n=2(a+pd-d)
½(m+n=a+(p-1)d
So pth term is ½(m+n)
QUESTION: 9
The first, second and last terms of an A.P. are a,b and 2a. The number of terms in the A.P. is :
Solution:
A.P : a , b , . . . . . . . . . . . . . .2a
1st term= a1 = a
2nd term = a2= b
nth term = an = 2a
d = a2 - a1 = b-a
an = a1 + (n-1)d = a + (n-1)(b-a) = 2a
(n-1)(b-a) = a
(n-1) = a / (b-a)
n = a / (b-a) + 1 = b / ( b -a )
Sn = n / 2 * ( a1 + an) = b / 2(b-a) * ( a + 2a) = 3ab / 2(b-a)
QUESTION: 10
Let s1, s2, s3 be the sums of n terms of three series in A.P., the first term of each being 1 and the common differences 1, 2, 3 respectively. If s1 + s3 = λs2, then the value of λ is :
Solution:
QUESTION: 11
Sum of first 5 terms of an A.P. is one fourth of the sum of next five terms. If the first term = 2, then the common difference of the A.P. is :
Solution:
QUESTION: 12
If x,y,z are in A.P., then the value of (x + y – z) (y + z – x) is equal to :
Solution:
QUESTION: 13
The number of numbers between 105 and 1000 which are divisible by 7 is :
Solution:
QUESTION: 14
If the numbers a,b,c,d,e form an A.P. then the value of a – 4b + 6c – 4d + e is equal to :
Solution:
QUESTION: 15
If sn denotes the sum of first n terms of an A.P., whose common difference is d, then sn – 2sn-1 + sn-2 (n >2) is equal to :
Solution:
sn−2sn−1+sn−2 = (sn−sn−1)−(sn−1−sn−2)
= an−an−1 [∵(sn−sn−1)= an]
= [a+(n−1)d]−[a+(n−2)d]
= a+nd−d−a−nd+2d
= d
QUESTION: 16
The sum of all 2-digited numbers which leave remainder 1 when divided by 3 is :
Solution:
The 2-digit number which when divided by 3 gives remainder 1 are: 10, 13, 16, ...97
Here a = 10, d = 13 - 10 = 3
tn = 97
nth term of an AP is tn = a + (n – 1)d
97 = 10 + (n – 1)3
⇒ 97 = 10 + 3n – 3
⇒ 97 = 7 + 3n
⇒ 3n = 97 – 7 = 90
Therefore, n = 90/3 = 30
Recall sum of n terms of AP,
Sn = n/2[2a + (n-1)d]
S30 = 30/2[2(10) + (30-1)3]
= 15[20 + 87] = 15 x 107 = 1605
Hence sum of 2-digit numbers which when divided by 3 yield 1 as remainder is 1605.
QUESTION: 17
The first term of an A.P. of consecutive integers is p2 + 1. The sum of 2p + 1 terms of this series can be expressed as :
Solution:
QUESTION: 18
If the sum of n terms of an AP is 2n2 + 5n, then its nth term is –
Solution:
QUESTION: 19
If the last term of an AP is 119 and the 8th term from the end is 91 then the common difference of the AP is –
Solution:
QUESTION: 20
If {an} = {2.5, 2.51, 2.52,...} and {bn} = {3.72, 3.73, 3.74,...} be two AP's, then a100005 – b100005 =
Solution:
Observing both the AP’s we see that the common difference of both the AP’s is same ,so difference between their corresponding terms will be same ie,a1-b1=2.5-3.72=-1.22
a2-b2=2.51-3.73=-1.22
So , a100005-b100005=-1.22
QUESTION: 21
If A1 and A2 be the two A.M.s between two numbers a and b, then (2A1 – A2) (2A2 – A1) is equal to :
Solution:
QUESTION: 22
are in A.P., then n is equal to
Solution:
QUESTION: 23
where Sn denotes the sum of the first n terms of an A.P., then the common difference of the A.P. is
Solution:
QUESTION: 24
If a,b,c are positive reals, then least value of (a + b + c) (1/a+1/b+1/c) is :
Solution:
QUESTION: 25
The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms.
Solution:<|endoftext|>
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# Year 4
### We can add and subtract mentally pairs of two-digit numbers and find a difference by counting on.We can make jottings to support mental calculations.
#### What we are learning:
• Addition can be done in any order. The order chosen should reflect the numbers in the addition. Always look for the simplest order to add the numbers together.
• Subtraction cannot be done in any order.
• When adding (mentally) a pair of 2-digit numbers the strategies that can be used include:
– Keep largest number whole, partition second number into tens and units. First add tens, then add units. i.e. to calculate 47+58, keep 58 whole then add 40 (=98) then add 7 (=105).
– Partition both numbers into tens and units. Add tens together, add units together then re-combine tens and units. i.e.to calculate 47 + 58, add tens 40+50 (=90) add units 7+8 (=15) then recombine tens and units 90+15 (=105).
– If one number is very ‘close’ to a multiple of 10, adjust the units so that it becomes a multiple of 10, then add second number. i.e. to calculate 47+58, 58 is close to 60. Take 2 units from 47 (leaving 45) and add them first to 58 (making 60). The new calculation is now 45+60.
• When subtracting (mentally) a pair of 2-digit numbers the strategies that can be used are more limited than for addition:
– Keep largest number whole, partition second number into tens and units. First subtract tens, then subtract units. i.e. to calculate 91 – 35, keep 91 whole then subtract 30 (=61) then subtract 5 this may need to be done in two stages as -1 then -4 (=56).
– An alternative method for subtraction is to find the difference between the two numbers by counting up from the lowest value to the highest value. i.e. to calculate 91-35, start on 35 add 5 (to get to 40) add 50 (to get to 90) add 1 (to get to 91). In total 56 has been added, so 91-35=56.
A jotting might be no more than writing down the numbers in the calculation and the ‘sub-totals’ calculated along the way. A jotting may also be a number line. A number line is a simple diagram (quick to draw – no need for scaling
accuracy) used to represent and support a calculation. It consists of a straight line (normally drawn horizontally). Key numbers should be added onto it.
##### ACTIVITY SHEETS
Activity sheet: Which Method PDF
#### Activities you can do at home:
Look together at the sheet “Which method?” Decide upon the best strategy to use to solve each calculation.
Use an Empty Number Line to support the calculations on “Which method?” See the example below:
91-47 = 44
#### Good questions to ask and discuss:
What types of jottings do you find most helpful?<|endoftext|>
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Home > Anthrax
Anthrax is a serious, sometimes deadly disease caused by infection with anthrax bacteria. These bacteria produce spores that can spread the infection.
Anthrax in humans is rare unless the spores are spread on purpose. It became a concern in the United States in 2001, when 22 cases occurred as a result of bioterrorism. Most of those cases affected postal workers and media employees who were exposed to spores when handling mail.
Most cases of anthrax occur in livestock, such as cattle, horses, sheep, and goats. Anthrax spores in the soil can infect animals who eat plants growing in the soil. People can be exposed to spores in infected animal products or meat. This is not much of a concern in North America, because livestock are vaccinated against anthrax. But people can get anthrax from handling animal skins or products made out of animal skins from parts of the world where anthrax is more common.
Anthrax is caused by Bacillus anthracis bacteria. There are three types of infection:
The illness does not seem to spread from person to person. People who come in contact with someone who has anthrax don't need to be immunized or treated unless they were exposed to the same source of infection.
The symptoms and the incubation period—the time from exposure to anthrax until symptoms start—depend on the type of infection you have.
With cutaneous anthrax, symptoms usually appear 5 to 7 days after exposure to spores, though it may take longer.
With inhalational anthrax, symptoms usually appear 1 to 7 days after exposure. (But it can take as long as 60 days).
With gastrointestinal anthrax, symptoms usually occur within a week after exposure.
Your doctor will ask you questions about your symptoms and about any work or other activities that may have put you at risk for exposure. If the doctor suspects you may have been exposed to anthrax, testing will be done to confirm exposure or infection. Public health officials also will be notified about a possible anthrax infection.
Anthrax is confirmed when the bacteria are identified from a culture of your blood, spinal fluid, skin sores, or mucus from your nose, airways, or lungs. If results of a culture aren't clear, you may need other blood tests or a polymerase chain reaction (PCR) test. A skin ulcer may be biopsied.
If your doctor thinks that you have inhalational anthrax, you may have a chest X-ray or a CT scan.
Antibiotics are used to treat all types of anthrax.
Anyone who is infected needs to be treated with antibiotics as soon as possible. Starting treatment before symptoms begin may make the illness less severe and prevent death. Treatment may also include supportive care in the hospital.
Anyone who has been exposed to anthrax spores but is not yet sick should be treated with antibiotics and a few doses of the vaccine to prevent infection. Not everyone who has been exposed to anthrax will get sick. But because there's no way to know who will get sick and who won't, anyone who is directly exposed will get treatment. If you think that you have been exposed, call your local law enforcement agency and your doctor right away. Don't take antibiotics without talking to your doctor first.
In the U.S., the anthrax vaccine is used to protect only the small number of people who are at higher risk for exposure. These include:
The vaccine is not available to the general public at this time. The risk of exposure to anthrax is extremely low.
The bioterrorism attacks in 2001 made many people nervous about opening their mail. If you receive a piece of mail that contains a powdery substance or seems suspicious, the U.S. Centers for Disease Control and Prevention (CDC) recommends that you put down the piece of mail and not touch it again. Then, leave the room, wash your hands with soap and water, and call 911 to find out what to do next.
If you have concerns about anthrax, you can find the most current information through the CDC (www.cdc.gov/anthrax).
Centers for Disease Control and Prevention (2001). Considerations for distinguishing influenza-like illness from inhalational anthrax. MMWR, 50(44): 985–987. Available online: http://www.cdc.gov/mmwr/preview/mmwrhtml/mm5044a5.htm.
Other Works Consulted
Duchin J, Malone JD (2009). Anthrax section of Bioterrorism. In EG Nabel, ed., ACP Medicine, section 8, chap. 5, pp. 8–16. Hamilton, ON: BC Decker.
Shadomy SV, Rosenstein NE (2008). Anthrax. In RB Wallace et al., eds., Wallace/Maxcy-Rosenau-Last Public Health and Preventive Medicine, 15th ed., pp. 1185–1194. New York: McGraw-Hill.
Current as ofJuly 30, 2018
Author: Healthwise StaffMedical Review: E. Gregory Thompson, MD - Internal MedicineAdam Husney, MD - Family MedicineKathleen Romito, MD - Family MedicineElizabeth T. Russo, MD - Internal MedicineLeslie A. Tengelsen, PhD, DVM - Epidemiology
Current as of:
July 30, 2018
Medical Review:E. Gregory Thompson, MD - Internal Medicine & Adam Husney, MD - Family Medicine & Kathleen Romito, MD - Family Medicine & Elizabeth T. Russo, MD - Internal Medicine & Leslie A. Tengelsen, PhD, DVM - Epidemiology
To learn more about Healthwise, visit Healthwise.org.
© 1995-2018 Healthwise, Incorporated. Healthwise, Healthwise for every health decision, and the Healthwise logo are trademarks of Healthwise, Incorporated.
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REQUEST FROM LAW ENFORCEMENT FOR RELEASE OF PROTECTED HEALTH INFORMATION 6051 US HIGHWAY 49, HATTIESBURG MS 39401 · 601-288-7000 · © FORREST HEALTH · ALL RIGHTS RESERVED ·<|endoftext|>
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## Engage NY Eureka Math 5th Grade Module 5 Lesson 1 Answer Key
### Eureka Math Grade 5 Module 5 Lesson 1 Problem Set Answer Key
Question 1.
Use your centimeter cubes to build the figures pictured below on centimeter grid paper. Find the total volume of each figure you built, and explain how you counted the cubic units. Be sure to include units.
Figure Volume Explanation A B C D E F
Question 2.
Build 2 different structures with the following volumes using your unit cubes. Then, draw one of the figures on the dot paper. One example has been drawn for you.
Question 3.
Joyce says that the figure below, made of 1 cm cubes, has a volume of 5 cubic centimeters.
a. Explain her mistake.
b. Imagine if Joyce adds to the second layer so the cubes completely cover the first layer in the figure above. What would be the volume of the new structure? Explain how you know.
a.
Joyce did not count the cube which is hidden. That is the cube on the second layer sitting on the hidden cube.
b.
I counted the first layer and multiplied by 2
5 x 2 = 10
Therefore, The volume of the new structure = 10cm3
### Eureka Math Grade 5 Module 5 Lesson 1 Exit Ticket Answer Key
Question 1.
What is the volume of the figures pictured below?
a.
5 cm3
b.
12cm3
Question 2.
Draw a picture of a figure with a volume of 3 cubic units on the dot paper.
### Eureka Math Grade 5 Module 5 Lesson 1 Homework Answer Key
Question 1.
The following solids are made up of 1 cm cubes. Find the total volume of each figure, and write it in the chart below.
Figure Volume Explanation A B C D E F
Question 2.
Draw a figure with the given volume on the dot paper.
Question 3.
John built and drew a structure that has a volume of 5 cubic centimeters. His little brother tells him he made a mistake because he only drew 4 cubes. Help John explain to his brother why his drawing is accurate.<|endoftext|>
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Maths Course in English
# Simple Interest Formulas: Short Tricks and Questions
Simple Interest Formulas: Short Tricks and Questions: In this page, you will learn about percentage Problems, Short Tricks in English. This topic is very important for every 2022 competitive exams preparation. So before proceeding for any exam read these short tricks and explanations carefully about Simple Interest Questions with Short Tricks with the explanation.
## Simple Interest with Example
Simple Interest is a fee paid by a borrower of assets to the owner as a form of compensation for the use of the assets. It is most commonly the price paid for the use of borrowed money or money earned by deposited funds.
When money is borrowed, interest is typically paid to the lender as a percentage of the principal, the amount owed to the lender. The percentage of the principal that is paid as a fee over a certain period of time (typically one month or year) is called the interest rate. A bank deposit will earn interest because the bank is paying for the use of the deposited funds. Assets that are sometimes lent with interest include money, shares, consumer goods through hire purchase, major assets such as aircraft, and even entire factories in finance lease arrangements. The interest is calculated upon the value of the assets in the same manner as upon money.
If the interest is paid as it falls due, at the end of the decided period (yearly, half yearly or quarterly) the principal is said to be lent or borrowed at simple interest.
### Quicker Method to solve the Questions
Simple Interest (SI)
Here P = principal, R = rate per annum, T = time in years
Therefore Amount (A)
If T is given in months, since rate is per annum, the time has to be converted into years, so the period in months has to be divided by 12. If T = 2 months this is years)
Example 1: Find the simple interest and amount when Rs. 1000 is lent at 5% per annum for 5 years.
Solution: By the formula,
SI =
= Rs. 60
∴ Amount = P + SI = 100 + 60 = Rs. 1060
Principal =
Example 2: Find the principal when simple interest is Rs. 60 at 4% per anum for 4 years.
Solution:
Principal =
= Rs. 750
Time =
Example 3: In how many years will the sum of Rs. 500 become Rs. 620 if the rate of simple interest is 4% per annum?
Solution: Using the formula,
Here, SI = 620 – 500 = Rs. 120
∴ = 6 years
Rate =
Example 4: At what rate percent per annum will a sum of money double in 8 years?
Solution: Let principal = Rs. P,Then SI = Rs. P and Time = 8 years
∴ Rate =
% per annum<|endoftext|>
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From 1776 to 1780 the words "State of Massachusetts Bay" appeared on the top of all acts and resolves. In 1780, the Massachusetts Constitution went into effect. Part Two of the Constitution, under the heading "Frame of Government" states: "that the people ... form themselves into a free, sovereign, and independent body politic, or state by the name of The Commonwealth of Massachusetts." Virginia (on June 29, 1776) and Pennsylvania (on September 25, 1776) adopted Constitutions which called their respective states commonwealths. Kentucky is also called a commonwealth in its full official state name (and in the Third Kentucky Constitution of 1850). Commonwealths are states, but the reverse is not true. The term "Commonwealth" does not describe or provide for any specific political status or legal relationship when used by a state. Those that do use it are equal to those that do not. Legally, Massachusetts is a commonwealth because the term is contained in the Constitution.
In the era leading to 1780, a popular term for a whole body of people constituting a nation or state (the body politic) was the word "Commonwealth." This term was the preferred usage of some political writers. There also may have been some anti-monarchial sentiment in using the word commonwealth. John Adams utilized this term when framing the Massachusetts Constitution.
Adams wrote: "There is, however, a peculiar sense in which the words republic, commonwealth, popular state, are used by English and French writers; who mean by them a democracy, or rather a representative democracy; a ‘government in one centre, and that centre the nation;’ that is to say, that centre a single assembly, chosen at stated periods by the people, and invested with the whole sovereignty, the whole legislative, executive, and judicial power, to be exercised in a body, or by committees, as they shall think proper." (Adams, John, and Charles Francis Adams. The Works of John Adams, Second President of the United States: with a Life of Author, Notes and Illustrations. Charles C. Little and James Brown, 1850-56, vol. 5, p. 454)
Partially adapted from the Secretary of the Commonwealth of Massachusetts<|endoftext|>
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# Inscribed Square Problem
Here’s another fun little geometry problem from MathChallenges.net. I found a different solution then what they gave. Let me know how you solve it.
## 3 thoughts on “Inscribed Square Problem”
1. This is a very intresting problem. We used to do some of these problems in Geometry with Mrs. Polishcuck.
—————————————————
What I did was…
So let’s name the sides of the square “s”.
and we know that a^2 and b^2 gets you the hypotunse.
We also know that the height of the mini top triangle is (a-s). and the width of the bottom right triangle is (b-s).
By finding and adding the hypotunse of these two mini triangles, you get the hypotunese of the big triangle.
In math terms…..
——————————————————
*Note: Sq( ) stands for square root.
sq(a^2 + b^2) = sq((a-s)^2 + s^2) + sq((b-s)^2 + s^2)
If you symplify it, you should get the answer….
I know that it gets really messy, and that this is probally not the most practical approach…but it should be correct.
Sorry that I didn’t have time to find out the final answer…since I had a english essay to finish. ^_^
but theoratically this approach should work despite the superlong squareroots and FOIL-ing that you have to do. XD
2. Spoiler Alert! Don’t read ahead if you don’t want to see the way I solved it.
I also solved it in a different way.
The way I solved it was by finding the area of the overall triangle, and setting it equal to the area of the sum of the small triangles and the square.
Labeling the sides of the square g I said:
(Attempting Latex, I hope it parses)
$\\ \frac{1}{2}ab = g^2 + \frac{1}{2}(b-g)(g) + \frac{1}{2}g(a-g) \\ ab = 2g^2 + (b-g)g + (a-g)g \\ ab = 2g^2 + (b + a - 2g)g \\ ab = 2g^2 + gb + ga - 2g^2 \\ ab = gb + ga \\ \frac{ab}{a+b} = g$
• Great! That’s how I did it too! I’m not sure what made me think to use area. And you’ll notice, if you look at their solution, they didn’t do it that way. They use a similar-triangles approach. Good work! 🙂
(And great work with the $LaTeX$ !)<|endoftext|>
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# If ‘-’ and ‘×’ signs as well as ‘7’ and ‘3’ are interchanged, then which of the one following is correct?A. 20 × 1 – 7 = 3B. 1 × 20 – 7 = 20C. 3 – 7 × 1 = 20D. 20 – 3 × 1 = 7
Free Practice With Testbook Mock Tests
## Options:
1. A
2. B
3. D
4. C
### Correct Answer: Option 4 (Solution Below)
This question was previously asked in
NTPC Tier I (Held On: 22 Apr 2016 Shift 2)
## Solution:
A. 20 × 1 – 7 = 3
After interchanging ‘-’ and ‘×’ signs as well as ‘7’ and ‘3’, we get:
20 – 1 × 3 = 7
L.H.S = 20 – 3
= 17 ≠ R.H.S
B. 1 × 20 – 7 = 20
After interchanging ‘-’ and ‘×’ signs as well as ‘7’ and ‘3’, we get:
1 – 20 × 3 = 20
L.H.S = 1 – 60
= -59 ≠ R.H.S
C. 3 – 7 × 1 = 20
After interchanging ‘-’ and ‘×’ signs as well as ‘7’ and ‘3’, we get:
7 × 3 – 1 = 20
L.H.S = 21 – 1
= 20 = R.H.S
D. 20 – 3 × 1 = 7
After interchanging ‘-’ and ‘×’ signs as well as ‘7’ and ‘3’, we get:
20 × 7 – 1 = 3
L.H.S = 140 – 1
= 139 ≠ R.H.S
Hence, ‘3 – 7 × 1 = 20’ is the correct answer.<|endoftext|>
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# Minecraft Calculus: River Crossing Escape
This is a classic introductory calculus problem, with a Minecraft theme. This particular variation is inspired by example 4 in section 4.5 of Stewart's calculus.
I'll set the scene: You're weaponless. You have half a heart and you're being chased by a skeleton archer. There's a river between you and your cabin. You need to get to your cabin as quickly as possible!
In more "mathy" terms, you want to get from $A$ to $B$. There are three main ways you can do this:
1. $A$ to $B$ directly, diagonally across the river
2. $A$ to $C$ and then $C$ to $B$
3. At slight angle from $A$ to $D$ and then $D$ to $B$
We know that swimming is slower than running, so option 1 isn't the best. Option 2 involves the smallest time in the water, but also the longest distance traveled.
Option 3 is somewhere in between - you spend a bit more time in the water, but the total distance is decreased. The travel time in this case depends on where exactly point $D$ is. To find the optimal path, we must find the position of $D$ which minimizes the travel time. This sounds like calculus!
Let $x$ be the distance between points $C$ and $D$. In math terms, $|CD|=x$.
We first need an expression for the total time traveled in terms of $x$. The basic equation for time traveled at constant speed is
$t=\frac{distance}{speed}$.
The first part of the trip is in the water where we travel from $A$ to $D$. We can use Pythagorean's theorem to get this distance as $|AD|=\sqrt{x^2+w^2}$. Assuming that we can travel at a speed of $v_w$ in the water, the time for this part is
$t_w=\frac{\sqrt{x^2+w^2}}{v_w}$.
The land part of the trip involves traveling what's left over of $l$ after having already traveled $x$, so $|DB|=l-x$. Traveling at $v_l$ on land, the time for this part of the trip is
$t_l=\frac{l-x}{v_l}$.
The total trip time, $T(x)$, is just the sum of these two.
$T(x)=\frac{\sqrt{x^2+w^2}}{v_w}+\frac{l-x}{v_l}$ (1)
To optimize this function with respect to $x$, we need to find where it is stationary and then verify that this point is a minimum. This means we want to find a place where the function isn't changing with small changes in $x$. In other words, we want to find a spot where the derivative is zero.
Taking the derivative gives:
$T'(x)=\frac{x}{v_w\sqrt{x^2+w^2}}-\frac{1}{v_l}$
Setting this equal to zero, we can solve for $x$:
$x=\frac{v_ww}{\sqrt{v_l^2-v_w^2}}$ (2)
Let's plug in some numbers! In Minecraft, you can swim at about 2.2 m/s and sprint at 5.6 m/s. Let's take the river width $w$ to be 7 blocks (1 block = 1 meter). Plugging in, we find that $x=2.9$. At this point, though, we can't tell if this is a maximum or a minimum. One way to find out is to examine the curvature of the function at this point by using the 2nd derivative:
$T''(x)=\frac{1}{v_w\sqrt{x^2+w^2}}(1-\frac{x^2}{x^2+w^2})$
Plugging in $x=2.9$, we see that $T''(2.9)=0.04$ which, being positive, means that $x=2.9$ is a minimum point for $T(x)$. Visually:
So if you want to get to your cabin as quickly as possible, the fastest route is to swim across the river to a point $D$ that is 2.9 meters from point $C$ and then run the rest of the way.
This is actually a general problem. Try replacing the speeds I used with speeds for soul sand, crouching and walking, a boat, etc. and see what happens! Particularly, what happens to (2) if you can travel more quickly in the water than on land?
#### Scratch Work
Finding $T'(x):$ First, rewrite the square root as a power, $T(x)=\frac{(x^2+w^2)^{\frac{1}{2}}}{v_w}+\frac{l-x}{v_l}$
Differentiating, using the chain rule on the first term, $T'(x)=\frac{\frac{1}{2}(x^2+w^2)^{-\frac{1}{2}}(2x)}{v_w}-\frac{1}{v_l}=\frac{x}{v_w\sqrt{x^2+w^2}}-\frac{1}{v_l}$
Finding $T''(x):$ Starting with $T'(x)=\frac{(x^2+w^2)^{-\frac{1}{2}}x}{v_w}+\frac{1}{v_l}$, we differentiate with respect to x. Note that the derivative of $\frac{1}{v_l}$ with respect to $x$ is 0, so we only have to deal with the first term. Using the product rule and chain rule:
$T''(x)=\frac{-\frac{1}{2}(x^2+w^2)^{-\frac{3}{2}}(2x^2)}{v_w}+\frac{(x^2+w^2)^{-\frac{1}{2}}}{v_w}$
Cleaning up with some algebra:
$T''(x)=\frac{-x^2}{v_w(x^2+w^2)^{\frac{3}{2}}}+\frac{1}{v_w\sqrt{x^2+w^2}} =\frac{1}{v_w\sqrt{x^2+w^2}}(1-\frac{x^2}{x^2+w^2})$<|endoftext|>
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## FORMULAS USED IN SET OPERATIONS
For any two finite sets A and B, we have the following useful results
(i) n(A) = n(A - B) + n(A n B)
(ii) n(B) = n(B - A) + n(A n B)
(iii) n(A U B) = n(A - B) + n(A n B) + n(B - A)
(iv) n(A U B) = n(A) + n(B) - n(A n B)
(v) n(A U B) = n(A) + n(B), when A n B = null set
(vi) n(A) + n(A') = n(U)
Example 1 :
If n(A n B) = 5, n(A U B) = 35, n (A) = 13, find n (B) .
Solution :
Using the formula,
n(A U B) = n(A) + n(B) - n(A n B)
35 = 13 + n (B) - 5
35 = 8 + n (B)
Subtract 8 on both sides
35 - 8 = 8 + n (B) - 8
27 = n (B)
Hence the value of n (B) is 27.
Example 2 :
If n (A) = 26, n (B) = 10, n (A U B) = 30, n (A') = 17, find n (A n B) and n (U) .
Solution :
Using the formula,
n(A U B) = n(A) + n(B) - n(A + B)
30 = 26 + 10 - n(A n B)
30 = 36 - n(A n B)
Subtract 36 on both sides
30 - 36 = 36 - n(A n B) - 36
-6 = - n(A n B)
Hence the value of n (A n B) is 6
n (U) = n (A) + n (A')
= 26 + 17 ==> n (U) = 43
Hence the value of n (U) is 43.
Example 3 :
If n(U) = 38, n(A) = 16, n(A n B) = 12, n(B') = 20
find n(A U B).
Solution :
n(A U B) = n(A) + n(B) - n(A n B)
In order to use the above formula to find n (AUB), we need the value of n (B).
n (B) + n (B') = n (U)
n (B) + 20 = 38
n (B) = 38 - 20 ==> 18
n(A U B) = 16 + 18 - 12
= 34 - 12
= 22
Hence the value of n(A U B) is 22.
Example 4 :
A and B are two sets such that n(A - B) = 32 + x, n(B - A) = 5x and n(A n B) = x Illustrate the information by means of a Venn diagram. Given that n(A) = n(B) . Calculate (i) the value of x (ii) n(A U B) .
Solution :
Formula to find n (A) is n(A) = n(A - B) + n(A n B)
Formula to find n (B) is n(B) = n(B - A) + n(A n B)
n(A) = 32 + x + x ==> 32 + 2x ---(1)
n(B) = 5x + x ==> 6x -----(2)
Given that n (A) = n (B)
32 + 2x = 6x
Subtract 2x on both sides
32 + 2x - 2x = 6x - 2x
32 = 4x
Divide by 4 on both sides, we get
32/4 = 4x /4
x = 8
(i) Hence the value of x is 8.
(ii) n(A U B) = n(A - B) + n(A n B) + n(B - A)
= 32 + x + x + 5x
= 32 + 7x
Applying the value of x,
= 32 + 7(8)
= 32 + 56
= 88
Kindly mail your feedback to [email protected]
## Recent Articles
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SAT Math Resources (Videos, Concepts, Worksheets and More)
2. ### SAT Math Videos (Part - 21)
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Index
# Discrete Mathematics Project Counting Techniques/Probability Activity
## Title
"What're the odds?" (Jim Arnow)
## Goals
1. Students learn to apply discrete counting techniques and concepts from discrete probability via an inherently interesting application -- the odds of various poker hands.
2. Students work in small groups to generate probability values then present their methods to the other members of the class.
## Abstract
This activity deals with computing probabilities of various poker hands. This topic is easily accessible to most students, since they may have played the game themselves, or are easily taught the game. As a group, the basics of calculating such probabilities are discussed with a specific hand, and then the students are left to apply the techniques to various other hands. This activity directly deals with the topic of discrete probability and counting techniques.
## Problem Statement
Discuss the game of poker with your students. The students should be made familiar with the various hands possible in poker as well as a small amount of some of the playing strategy. Included in this strategy should be a discussion of the fact that in poker it is useful if a player knows which hands are more common than others. The more you know, the easier it is to make decisions about various hands. To determine which hands are more common, we can use counting techniques to determine their relative frequencies.
## Instructor Suggestions
Possible lesson plan:
1. Discuss the rules until the students are familiar with how the game is played. If there are some students who are familiar with the rules of the game, have them present them to the whole class.
2. For simplicity, the situation should be restricted to a game consisting of five cards with no draw.
3. Once the students are comfortable with the rules of the game, discuss the number of different possible hands that can be drawn with five cards. This can be presented as an example of the multiplication principle or as a combination. (The total number of possible hands is C(52,5) or 2,598,960.)
4. Discuss the definition of probability in terms of counting -- the number of possible ways of an outcome occurring divided by the total number of outcomes.
5. Calculate, as a large group, the number of possible ways of drawing a representative hand. A full house (total number -- 3,744) and/or a straight flush (total number -- 40) are good examples, since they include some features of other hands.
6. Divide the class into small groups and give each group the task of calculating the number of possible ways of drawing two or three of the possible hands. If possible, it may be best to assign hands to more than one group so that they may compare answers.
7. When all the groups have made sufficient progress, have a spokesperson for each group present their results to the class. Discuss the results as a group and compare the results with the results published in Hoyle or similar books on card games.
Notes:
• These calculations can be very confusing and even intimidating to students. To make them easier to approach, have the students describe various hands in terms of their distinct characteristics. For instance, to uniquely determine a straight flush, we need only determine the lowest card (ten possible ways), and the suit (four possible ways). The total number of straight flushes is then found by the multiplication principle -- 4 times 10 or 40.
• Students can get very confused by the fact that the cards are dealt sequentially. Remind them that a hand is the same regardless of what order the cards are dealt -- we can always sort the cards in our hands and that doesn't affect what the hand is.
• Some useful terms from poker:
Rank:
The numerical value of the card, Ace, 2, 3, 4, 5, 6, 7, 8, 8, 10, Jack, Queen or King.
Suit:
Hearts (H), Clubs (C), Diamonds (D) or Spades (S)
• How to make the various calculations (other explanations are possible):
Straight flush:
4 ways of choosing the suit and ten choices for low (or high) card. Total -- 40 ways.
Four of a kind:
13 choices for the rank, 48 choices for the other card. Total -- 624 ways.
Full House:
Thirteen choices for the rank of the three of a kind, C(4,3) ways to choose three cards of that rank, 12 ways to choose the rank of the pair, C(4,2) ways to choose two cards of that rank. Total -- 3,744 ways.
Flush:
Four choices for the suit, C(13,4) choices for the four cards minus 40 straight flushes which have already been counted. Total -- 5108 ways.
Straight:
Ten choices for the rank of the lowest card times 4 for the choice for the suit of each card minus 40 for the straight flushes which have already been counted. Total 10,200 ways.
Three of a kind:
13 choices for the rank of the three of a kind, C(4,3) ways to choose three cards of that rank, 48 choices for one of the remaining cards, 44 choices for the other (not 47 since it can't be the same rank as the other non-matching card), divide by two for the possible permutations of the two non-matching cards. Total -- 54,912 ways.
Two Pairs:
C(13,2) ways to choose the ranks of the two pairs, C(4,2) ways to choose which cards of the first rank, C(4,2) ways to choose which cards of the second rank and 44 choices for the remaining card. Total -- 123,553 ways.
One Pair:
13 ways to choose the rank of the pair, C(4,2) ways to choose which cards from that rank, 48 choices for the first non-matching card, 44 choices for the next non-matching card, 40 choices for the last non-matching card, divide by P(3,3) or the possible permutations of the three non-matching cards. Total ways -- 1,098,240.
• If the situation arises, the difference between odds and probability can be discussed. Remember that the probability of an event occuring can be calculated as the number of ways that the event can occur divided by the total number of outcomes. The odds of something occuring can be calculated as the ratio of the number of ways that the event can occur to the number of ways in which the event does not occur.
## Materials
decks of cards, "What're the odds?" handout, handout of Hoyle's count of poker hands, calculators, scratch paper
## Time
Discussion of rules of poker (5 min.), discussion of how to calculate count for one example (10 min.), Small group work (30 min. to 1 hour), presentation of small group work and large group discussion (15 min.)
## Mathematics Concepts
### Discrete Mathematics
Counting techniques, probability, permutations and combinations, odds and probability, independent events
### Related Mathematics
factorial, sequences, sets, common factors (for reducing probability calculations)
## NCTM Standards Addressed
Problem Solving, Communication, Reasoning, Connections, Algebra, Probability
## Colorado Model Content Standards Addressed
Number Sense (1), Algebraic Methods (2), Data Collection and Analysis (3), Problem Solving Techniques (5), Linking Concepts and Procedures (6)
## Curriculum Integration
This activity could be integrated:
1. into an algebra class which has covered the definition of factorial
2. into a probability or statistics class as an exposure to a classic "real world" application of probability
## Further Investigation
The concept of calculating probabilities can be extended to a number of similar games, such as roulette, craps, dungeons and dragons, etc. The students should be able to come up with a variety of possible applications.
Some related questions:
1. What is the probability of getting one or the other of some of these hands, e.g. What is the probability of drawing a flush or four of a kind?
2. What are some important probabilities if I am playing five card poker where I am allowed to draw? (i.e. I can return any number of my cards and replace them from the deck.)
3. Do my probabilities change if I am playing with a number of other people and they are dealt their cards before I am? (A: No) Why not?
4. What would change about this problem if we were to draw seven cards and take our best five from them? How could we approach the problem?
5. What other games have probabilities such as this to be calculated?
## References, Resources and Related Web Links
Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete Mathematics Through Applications. New York: W. H. Freeman and Company.
Frey, Richard L. (1964). The New Complete Hoyle. New York: Doubleday.
Ross, S. (1984). A First Course in Probability. New York: Macmillan Publishing.<|endoftext|>
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# Using Euclid’S Algorithm, Find the Hcf Of 960 and 1575 . - Mathematics
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Using Euclid’s algorithm, find the HCF of 960 and 1575 .
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#### Solution
On applying Euclid’s algorithm, i.e. dividing 1575 by 960, we get:
Quotient = 1, Remainder = 615
∴ 1575 = 960 × 1 + 615
Again on applying Euclid’s algorithm, i.e. dividing 960 by 615, we get:
Quotient = 1, Remainder = 345
∴ 960 = 615 × 1 + 345
Again on applying Euclid’s algorithm, i.e. dividing 615 by 345, we get:
Quotient = 1, Remainder = 270
∴ 615 = 345 × 1 + 270
Again on applying Euclid’s algorithm, i.e. dividing 345 by 270, we get:
Quotient = 1, Remainder = 75
∴ 345 = 270 × 1 + 75
Again on applying Euclid’s algorithm, i.e. dividing 270 by 75, we get:
Quotient = 3, Remainder = 45
∴ 270 = 75 × 3 + 45
Again on applying Euclid’s algorithm, i.e. dividing 75 by 45, we get:
Quotient = 1, Remainder = 30
∴ 75 = 45 × 1 + 30
Again on applying Euclid’s algorithm, i.e. dividing 45 by 30, we get:
Quotient = 1, Remainder = 15
∴ 45 = 30 × 1 + 15
Again on applying Euclid’s algorithm, i.e. dividing 30 by 15, we get:
Quotient = 2, Remainder = 0
∴ 30 = 15 × 2 + 0
Hence, the HCF of 960 and 1575 is 15.
Concept: Euclid’s Division Lemma
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Discovering the answers to two basic questions—primarily, “How do I recognize evidence of geologic change in my environment?” and secondarily, “What is the source of the Crissy Field riprap rocks?”—is the ultimate goal of the field-based Rockin’ in the Riprap experience. In order to achieve this goal, participants use two data sheets, Riprap Rocks—Field Observations and Crissy Field Center Observations and Notes, to record what they see and think. The information sheets, which include photos of various riprap rock types and maps, provide information not accessible at the site. (For more on the decision to use the riprap location, see Planning a Place-based Inquiry Experience for Teachers.)
The essential and secondary questions appear in the upper right-hand corner of each sheet. Recognizing that, similar to school-age students, adults can lose track of an investigation’s main question, we decided to keep the question(s) in front of them throughout. And, because a wealth of information can be gleaned and organized in this field-based experience, participants are asked to write down their observations. The second data sheet suggests questions to prompt the development of explanatory theories in a small-group context prior to discussing the points with the group as a whole.
The Riprap Rocks—Field Observations data sheet is used to record descriptive observations of the half-dozen most important rocks found in the riprap (we do a bit of seeding, which we acknowledge, to ensure that these six rock types are represented). The one-page information sheet, Describing Rocks, illustrates the characteristics geologists use to identify rocks, including grain size, veining, and matrix material. (Students are asked to ignore asphalt and concrete—good recycling, but not relevant to this investigation.) We also carefully select the quadrants each group uses and prompt them when necessary to include particular rocks.
The second data sheet, Crissy Field Center Observations and Notes, guides the participants through four tasks. First, they match their field observations to a set of six labeled rock photographs as well as to samples of these rocks. Second, with names of the riprap rocks in hand, participants review information sheets on each of the rock types: Pillow Basalt, Serpentinite, Radiolarian Chert, Graywacke Sandstone, Granite, and Andecite and Dacite. These information sheets provide more views of the rocks (including maps, if the rocks are not from the Marin or San Francisco headlands), and very brief descriptions (roughly 50 to 75 words) of the conditions under which the rocks were formed. Third, participants examine two maps of the headlands to locate local outcroppings of the four Franciscan Complex rocks. They also examine photographs of Cherts and Pillow Basalts from Pacific Rim locations (Alaska, Japan, Washington state, British Columbia, and Malaysia). Finally, to prompt critical thinking about their observations, they respond to questions on the second data sheet: What is the local source of the riprap rocks? What are their sites of origin? What geologic process could account for their current locations? (Participants with significant geology backgrounds are segregated so they do not give away the answers, thereby spoiling the inquiry experience for the others.)
This segment follows the guidelines for inquiry investigations outlined in Inquiry and the National Science Educations Standards (National Research Council 2000). Participants are engaged in a scientifically oriented question, they give priority to evidence, they formulate explanations based on evidence, and there is a connection to scientific knowledge. These essential features are outlined in an inquiry grid that is reproduced here (the volume is available on-line at http://books.nap.edu/catalog).
Last updated: February 28, 2015<|endoftext|>
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Sunday, August 14, 2022
# Algebra 1 PARCC question: area of rug equation
-
#### The following constructed response question, explained here in hopes of helping algebra students in Maryland and Illinois prepare for the PARCC test near the end of this school year, appears on the released version of PARCC’s Spring 2015 test in algebra 1, here:
Tonya has a rectangular rug with an area of 21 square feet. The rug is 4 feet longer than it is wide.
Part A
Create an equation that can be used to determine the length and width of the rug. Justify your answer.
Part B
Tonya adds a 1.5-foot border all the way around the rug. What is the area of the enlarged rug? Show all your work.
Correct answers: Part A: You can use the system l = w – 4 and l × w = 21. Part B: 60 ft2. Both Part A and Part B are human-scored.
PARCC evidence statement(s) tested: HS.D.2-5:
Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of course-level knowledge and skills articulated in A-CED, N-Q, A-SSE.3, A-REI.6, A-REI.12, A-REI.11-1, limited to linear equations and exponential equations with integer exponents.
A-CED (A-CED.4-2, e.g., rearrange formulas that are quadratic in the quantity of interest to highlight the quantity of interest, using the same reasoning as in solving equations) is the primary content; other listed content elements may be involved in tasks as well.
The evidence statement above references Math Practice 2 and Math Practice 4 in the Common Core:
[2] Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
[4] Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. … By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
The question tests students’ understanding of the high school Common Core algebra standard HSA.CED.A.1, found under high school algebra (creating equations), which states that they should be able to “Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.”
Example of a solution strategy (there are others)
Part A: Use the area formula for a rectangle, A = w × l.
Set the longer dimension of the rectangle to the variable w, the area of the rectangle to 21, and the shorter side of the rectangle to w – 4.
$\begin{array}{rcl} A = w (w-4) & = & 21 \\ w^2 - 4w - 21 & = & 0 \end{array}$
I can factor the left side by noting that –7 × 3 = –21 and –7 + 3 = –4.
$(w-7)(w+3) = 0$
Once we get to this point, we know from the multiplicative property of 0 that if a × b = 0, then either a = 0, b = 0, or both a and b = 0.
If (w–7) = 0, then w is 7, and if (w+3) = 0, then w is –3. The width can’t be less than zero, so the –3 isn’t useful in the context of the problem. That leaves a width of 7 feet. The other dimension is 4 feet less than that, so 3 feet.
Part B: Use the area formula for a rectangle, A = w × l.
The new dimension of the rug, with a 1.5-foot border added on each of the four sides is (7 + 1.5 + 1.5) feet by (3 + 1.5 + 1.5) feet.
$A = (10)(6) = 60$
## Resources for further study
Purple Math, developed by Elizabeth Stapel, a math teacher from the St Louis area, has a six-part series on solving quadratic equations by factoring, as I did above. The series starts here.
The Khan Academy, developed by Sal Khan, an engineer who has created a library of thousands of video lessons, has a series of video lessons that demonstrate how to factor quadratics, starting here. He starts by factoring x2 – 14x + 40 as (x–4)(x–10), even though the text on his site says the second factor is (x–1). Despite the minor error in marketing, which would be unforgivable if done on the PARCC test, the video’s fine. Mr Khan clearly understands completely how to factor quadratics, even if he typed it incorrectly on his site.
Chapter 4, Section 4.4, of the book Algebra 2, Illinois edition by Ron Larson et al deals with solving quadratic equations by factoring. Students are trained to spot the structure in polynomial expressions of the general form
$ax^2 + bx + c = (kx+m)(lx+n) = klx^2 + (kn+lm)x + mn$
where k and l must be factors of a and m and n must be factors of c.
Complete reference: Ron Larson, Laurie Boswell, Timothy D Kanold, Lee Stiff. Algebra 2, Illinois edition. Evanston, Ill.: McDougal Littell, a division of Houghton Mifflin Company, 2008. The book is used in several algebra classes taught in Illinois high schools.
## Analysis of this question and online accessibility
The question measures knowledge of the Common Core standard it purports to measure and tests students’ ability to create an equation to model a real-world situation. The knowledge of the area of a rectangle is both given to students on a formula sheet and considered securely-held knowledge, and so that is not tested here, just used. It is considered to have a median cognitive demand.
(We note that most ninth graders can probably figure out what the length and width of the original rug is in their heads. What number, when multiplied by a number that is 4 less, gives 21? That’s a fairly simple logic they can do by trial and error or “guess and check.” Try 6: 6 × 2 = 12; nope. Try 7: 7 × 3 = 21; bingo! But Part A assesses the Common Core skill cited here, which is students’ ability to construct a mathematical model for a real-world situation. Just getting the 7 and the 3 will earn a point out of 6 total given to this problem. The bulk of the points come from the mathematical modeling—coming up with the equation.)
The question can be tested online and should yield results that are as valid and reliable as those obtained on paper. Students online may experience difficulties with the equation editor, as the use of this online tool is required to receive full credit in both Parts A and B.
If students are unfamiliar with the tool—which requires them to enter math work in paragraph form by selecting math symbols from a series of drop-down palettes and does not in any way resemble the way they would do the work if given a pencil and paper—their score will be in jeopardy. Typos are not forgiven in the PARCC scoring rubrics, and students are advised to take a little extra time when using the equation editor tool to make sure they have
• Entered all the work or logic necessary
• Transferred all work from scratch paper to the computer
(I realize using the equation editor is difficult, but you’re not alone. And while people at PARCC are trying to figure out how to make this tool usable for an online test, that’s not going to happen anytime soon. If PARCC people were here, they would be apologizing profusely for this poorly conceived online monstrosity. But they’re not; you’re here, and you have to do this test if you live in a PARCC state.)
The Maryland and Illinois governments have passed laws giving the respective state boards of education the responsibility of adopting standards of learning for students. Good or bad, both states have adopted customized versions of the Common Core, which, in math, incorporates what are known as “math practices.”
The principles are considered overarching. In spirit, then, no one solution strategy is endorsed or disallowed, but students are expected to be creative and use whatever tool or solution strategy they feel is appropriate to solve a given math problem. This question makes it very difficult for students to achieve full credit if they solved Part B by making a drawing of the rug with the border, a perfectly valid and plausible solution strategy for the question as asked. The equation editor online doesn’t allow them to show this work and therefore robs them of point-earning potential in the scoring, especially in Part B.
This clearly violates the Common Core’s prevailing math principles, which both states have duly adopted by willful action of the appropriate governing bodies.
(The use of technology to show “all” work, in paragraph form, is an artifact of PARCC and results from the consortium’s interpretation of and self-imposed restrictions on what the Common Core math practices refer to as the use of technology. This artificial limitation imposed by the test penalizes creative students, who are unable to show their work on this problem in a way that would be appropriate to the task at hand: area. The question, as delivered and scored online, does not align to several key aspects of the Common Core and is therefore invalid. On paper, the question is completely valid, though.)
Note that the writing tasks on the PARCC test and especially the various multiple-choice formats, despite being enhanced by technology, fail to represent the rigor required by the Common Core, as Maryland and Illinois have adopted the standards. The Hechinger Report had this to say:
The Common Core tests contain multiple-choice questions and some writing tasks that don’t measure up to the ambitious Common Core education goals with which they are supposed to be aligned. … If students are taught to question what they’ve learned and reflect on the source of their knowledge, why should they be judged by a test on which they must choose from among several pre-fab answers?
No special accommodation challenges can be identified with this question, so the question is considered fair.
## Challenge
Explain why, using the form of a two-column mathematical proof, the quadratic formula would yield exactly the same solutions as the factoring method used above. For reference the quadratic formula is
$x = \frac{b \pm \sqrt{b^2 - 4ac}}{2a}$
where a is the coefficient of x2, b is the coefficient of x, and c is the constant in a quadratic equation of the form
$ax^2 + bx + c = 0$
## Purpose of this series of posts
Voxitatis is developing blog posts that address every algebra 1 question released to the public by the Partnership for Assessment of Readiness for College and Careers, or PARCC, in order to help students prepare to take the test this spring.
Our total release will run from February 27 through March 15, with one or two questions discussed per day. Then we’ll move to geometry at the end of March, algebra 2 during the first half of April, and eighth grade during the last half of April.
Paul Katulahttps://news.schoolsdo.org
Paul Katula is the executive editor of the Voxitatis Research Foundation, which publishes this blog. For more information, see the About page.
### In Kennedy v Bremerton, Lemon finally falls
0
A Supreme Court decision Monday eroded, to a certain extent, the proverbial wall between church and state.<|endoftext|>
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14-2 Frequency and Histograms
```Frequency
Frequencyand
andHistograms
Histograms
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Holt
McDougal
Algebra 1Algebra
Algebra11
Holt
McDougal
Frequency and Histograms
Warm Up
Identify the least and greatest value in
each data set.
1. 34, 62, 45, 35, 75, 23, 35, 65, 23 23, 75
2. 1.6, 3.4, 2.6, 4.8, 1.3, 3.5, 4.0 1.3, 4.8
Order the data from least to greatest.
3. 2.4, 5.1, 3.7, 2.1, 3.6, 4.0, 2.9
2.1, 2.4, 2.9, 3.6, 3.7, 4.0, 5.1
4. 5, 5, 6, 8, 7, 4, 6, 5, 9, 3, 6, 6, 9
3, 4, 5, 5, 5, 6, 6, 6, 6, 7, 8, 9, 9
Holt McDougal Algebra 1
Frequency and Histograms
Objectives
Create stem-and-leaf plots.
Create frequency tables and
histograms.
Holt McDougal Algebra 1
Frequency and Histograms
Vocabulary
stem-and-leaf plot
frequency
frequency table
histogram
cumulative frequency
Holt McDougal Algebra 1
Frequency and Histograms
A stem-and-leaf plot arranges data by dividing
each data value into two parts. This allows you
to see each data value.
The digits other than
the last digit of each
value are called a
stem.
The last digit of a
value is called a leaf.
Key: 2|3 means 23
The key tells you how
to read each value.
Holt McDougal Algebra 1
Frequency and Histograms
Example 1A: Making a Stem-and-Leaf Plot
The numbers of defective widgets in batches
of 1000 are given below. Use the data to
make a stem-and-leaf plot.
14, 12, 8, 9, 13, 20, 15, 9, 21, 8, 13, 19
Number of Defective
Widgets per Batch
Stem Leaves
0
1
2
8899
233459
01
Key: 1|9 means 19
Holt McDougal Algebra 1
The tens digits are the stems.
The ones digits are the
leaves. List the leaves
from least to greatest
within each row.
Title the graph and add a key.
Frequency and Histograms
Example 1B: Making a Stem-and-Leaf Plot
The season’s scores for the football teams
going to the state championship are given
below. Use the data to make a back-to-back
stem-and-leaf plot.
Team A: 65, 42, 56, 49, 58, 42, 61, 55, 45, 72
Team B: 57, 60, 48, 49, 52, 61, 58, 37, 63, 48
Holt McDougal Algebra 1
Frequency and Histograms
Example 1B Continued
Team A: 65, 42, 56, 49, 58, 42, 61, 55, 45, 72
Team B: 57, 60, 48, 49, 52, 61, 58, 37, 63, 48
Football State
Championship Scores
Team A
3
9522 4
865 5
51 6
2 7
Team B
7
889
278
013
Key: |4|8 means 48
2|4| means 42
Holt McDougal Algebra 1
The tens digits are the stems.
The ones digits are the
leaves.
Put Team A’s scores on the
left side and Team B’s
scores on the right.
Title the graph and add a key.
Frequency and Histograms
Check It Out! Example 1
The temperature in degrees Celsius for two
weeks are given below. Use the data to make a
stem-and-leaf plot.
7, 32, 34, 31, 26, 27, 23, 19, 22, 29, 30, 36, 35, 31
Temperature in
Degrees Celsius
Stem
0
1
2
3
Key: 1|9
Leaves
7
9
23679
0112456
means 19
Holt McDougal Algebra 1
The tens digits are the stems.
The ones digits are the
leaves. List the leaves
from least to greatest
within each row.
Title the graph and add a key.
Frequency and Histograms
The frequency of a data value is the number of
times it occurs. A frequency table shows the
frequency of each data value. If the data is
divided into intervals, the table shows the
frequency of each interval.
Holt McDougal Algebra 1
Frequency and Histograms
Example 2: Making a Frequency Table
The numbers of students enrolled in Western
Civilization classes at a university are given
below. Use the data to make a frequency table
with intervals.
12, 22, 18, 9, 25, 31, 28, 19, 22, 27, 32, 14
Step 1 Identify the least and greatest values.
The least value is 9. The greatest value is 32.
Holt McDougal Algebra 1
Frequency and Histograms
Example 2 Continued
Step 2 Divide the data into equal intervals.
For this data set, use an
interval of 10.
Step 3 List the intervals
in the first column of the
table. Count the number
of data values in each
interval and list the count
in the last column. Give
the table a title.
Holt McDougal Algebra 1
Enrollment in Western
Civilization Classes
Number
Frequency
Enrolled
1 – 10
1
11 – 20
21 – 30
4
5
31 – 40
2
Frequency and Histograms
Check It Out! Example 2
The number of days of Maria’s last 15
vacations are listed below. Use the data to
make a frequency table with intervals.
4, 8, 6, 7, 5, 4, 10, 6, 7, 14, 12, 8, 10, 15, 12
Step 1 Identify the least and greatest values.
The least value is 4. The greatest value is 15.
Step 2 Divide the data into equal intervals.
For this data set use an interval of 3.
Holt McDougal Algebra 1
Frequency and Histograms
Check It Out! Example 2 Continued
Step 3 List the intervals in the first column of
the table. Count the number of data values in
each interval and list the count in the last
column. Give the table a title.
Number of Vacation Days
Holt McDougal Algebra 1
Interval
Frequency
4–6
5
7–9
4
10 – 12
4
13 – 15
2
Frequency and Histograms
A histogram is a bar graph used to display
the frequency of data divided into equal
intervals. The bars must be of equal width
and should touch, but not overlap.
Holt McDougal Algebra 1
Frequency and Histograms
Example 3: Making a Histogram
Use the frequency table in Example 2 to
make a histogram.
Step 1 Use the scale and
interval from the frequency
table.
Step 2 Draw a bar for the
number of classes in each
interval.
All bars should be the
same width. The bars
should touch, but not
overlap.
Holt McDougal Algebra 1
Enrollment in Western
Civilization Classes
Number
Frequency
Enrolled
1 – 10
1
11 – 20
21 – 30
4
5
31 – 40
2
Frequency and Histograms
Example 3 Continued
Step 3 Title the graph
and label the horizontal
and vertical scales.
Holt McDougal Algebra 1
Frequency and Histograms
Check It Out! Example 3
Make a histogram for the number of days of
Maria’s last 15 vacations.
4, 8, 6, 7, 5, 4, 10, 6, 7, 14, 12, 8, 10, 15, 12
Step 1 Use the scale and interval from the
frequency table.
Number of Vacation Days
Holt McDougal Algebra 1
Interval
Frequency
4–6
7–9
5
4
10 – 12
13 – 15
4
2
Frequency and Histograms
Check It Out! Example 3 Continued
Step 2 Draw a bar for the number of scores in
each interval.
All bars should be the
same width. The bars
should touch, but not
overlap.
Step 3 Title the graph
and label the horizontal
and vertical scales.
Holt McDougal Algebra 1
Vacations
Frequency and Histograms
Cumulative frequency shows the frequency of
all data values less than or equal to a given
value. You could just count the number of
values, but if the data set has many values, you
might lose track. Recording the data in a
cumulative frequency table can help you keep
track of the data values as you count.
Holt McDougal Algebra 1
Frequency and Histograms
Example 4: Making a Cumulative Frequency Table
The weights (in ounces) of packages of
cheddar cheese are given below.
19, 20, 26, 18, 25, 29, 18, 18, 22, 24, 27, 26,
24, 21, 29, 19
a. Use the data to make a cumulative
frequency table.
Step 1 Choose intervals for the first column of the
table.
Step 2 Record the frequency values in each interval
for the second column.
Holt McDougal Algebra 1
Frequency and Histograms
Example 4 Continued
Step 3 Add the frequency of each interval to the
frequencies of all the intervals before it. Put that
number in the third column of the table.
Step 4 Title the
table.
Holt McDougal Algebra 1
Cheddar Cheese
Weight
(oz)
Frequency
Cumulative
Frequency
18-20
6
6
21-23
2
8
24-26
5
13
27-29
3
16
Frequency and Histograms
Example 4 Continued
b. How many packages weigh less than 24 ounces.
All packages less
than 24 oz are
displayed in the first
two rows of the
table, so look at the
cumulative frequency
shown in the second
row.
There are 8 packages
with weights under
24 oz?
Holt McDougal Algebra 1
Cheddar Cheese
Weight
(oz)
Frequency
Cumulative
Frequency
18-20
6
6
21-23
2
8
24-26
5
13
27-29
3
16
Frequency and Histograms
Check It Out! Example 4
The number of vowels in each sentence of a
short essay are listed below.
33, 36, 39, 37, 34, 35, 43, 35, 28, 32, 36,
35, 29, 40, 33, 41, 37
a. Use the data to make a cumulative
frequency table.
Step 1 Choose intervals for the first column of the
table.
Step 2 Record the frequency values in each interval
for the second column.
Holt McDougal Algebra 1
Frequency and Histograms
Check It Out! Example 4 Continued
Step 3 Add the frequency of each interval to the
frequencies of all the intervals before it. Put that
number in the third column of the table.
Step 4 Title the
table.
Holt McDougal Algebra 1
Vowels in Sentences
Number
Frequency
Cumulative
Frequency
28-31
2
2
32-35
7
9
36-39
5
14
40-43
3
17
Frequency and Histograms
Check It Out! Example 4 Continued
b. How many sentences contain 35 vowels or fewer?
All sentences with
less than 35 vowels.
are displayed in the
first two rows of the
table, so look at the
cumulative frequency
shown in the second
row.
There are 9 sentences
with fewer than 35
vowels.
Holt McDougal Algebra 1
Vowels in Sentences
Number
Frequency
Cumulative
Frequency
28-31
2
2
32-35
7
9
36-39
5
14
40-43
3
17
Frequency and Histograms
Lesson Quiz: Part I
1. The number of miles on the new cars in a car
lot are given below. Use the data to make a
stem-and-leaf plot.
35, 21, 15, 51, 39, 41, 46, 22, 28, 16, 12, 40,
34, 56, 25, 14
Holt McDougal Algebra 1
Frequency and Histograms
Lesson Quiz: Part II
2. The numbers of pounds of laundry in the
washers at a laundromat are given below. Use
the data to make a cumulative frequency table.
2, 12, 4, 8, 5, 8, 11, 3, 6, 9, 8
Holt McDougal Algebra 1
Frequency and Histograms
Lesson Quiz: Part III
3. Use the frequency table from Problem 2 to
make a histogram.
Holt McDougal Algebra 1
```<|endoftext|>
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Master the 7 pillars of school success
Many times after adding fractions,multiplying fractions, or dividing fractions, you are left with a fraction that needs to be simplified or reduced.
• This method of simplifying a fraction involves using a factor tree in order to reduce the fraction.
• Here are the steps for reducing or simplying a fraction using a factor tree.
factor tree is a diagram that helps you find the prime factors of a number. You find the factor of a number, and then the factors of that number until you can’t factor anymore. The end result of the factor tree is all the prime factors of the original number.
Step 1. Make a factor tree for the top number ( numerator) and the bottom number ( denominator).
Step 2. Underline prime numbers on your factor tree.
Step 3. Make a list of the prime numbers for your numerator and denominator.
Step 4. Cancel common factors found on the numerator and the denominator
Step 5. Multiply the remaining factors.
Let's look at an example from the video:
Simplify or Reduce the fraction 16
120
# Simplifying Fractions
Step 1. Set up a factor tree for your numerator and your denominator Your numerator is the top number of your fraction and denominator is the bottom number
Step 2. Underline the prime numbers on both factor trees. Prime numbers are only divisible by itself and 1
Step 3. Make a list of the prime numbers for your numerator and your denominator
Step 5. Multiply any remaining factors found in the numerator and denominator.
Step 4. Cross out common factors.There is not a 5 or 3 on the numerator so they are not crossed out.
## Definition of a Complex Fraction
A complex fraction is a fraction with a fraction within a fraction. In other words the denominator,numerator,or both contain a fraction.
## How do you simplify a Complex Fraction ?
I feel the easiest method for simplifying a complex fraction is to rewrite the fraction as a division problem by using, Change,Keep,Flip
Example 1 Simplify the following Complex Fraction
Lets look at the parts of this complex fraction
Numerator
You can think of the vinculum as a division sign
Denominator
Step 1 Simplify the numerators if you can. In this problem add 5+7=12 and then change the fraction into a division problem. See step 2.
Step 2 Rewrite the division problem to a multiplication problem by using the "Keep,Change,Flip Method"
Keep the first fraction the same
Change the sign
Flip the last fraction/ Create a reciprocal
Step 3 Multiply straight across in order to get the complex fraction in the simplest form
## Simplify a complex fraction that includes a variable
Step 1.
Step 2.
Step 3.
x
x
Simplify this complex fraction that has an exponent
Simplify the numerator using the common denominator 2x
Rewrite the equation using Keep,Change,Flip
Keep the first fraction the same
Change the sign
Flip the last fraction/ Create a reciprocal
Step 4 Multiply straight across in order to get the complex fraction in the simplest form
How do you simplify a complex question?
What is a complex fraction?
How do you simplify 5+7/11/5/12?
How do you simplify x/2 +1/x/x^2 +2x?
Simplifying Complex Fractions
Common Core Standard: 4.NF.1
To simplify a fraction means to reduce a fraction to its lowest terms.
A fraction is in lowest terms when the numerator and denominator have no common factor other than one.<|endoftext|>
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Knowledge and understanding of current events are an important part of being a good citizen. As adults, most of use dislike the feeling that there is something going on in the world that we know nothing about. Wouldn't it be great if kids would have that same feeling and be driven to know the current events of the world in which they live? Through an account with Discovery Education, you have access to the Global Wrap, a resource that will allow you to instill that drive in your students. Global Wrap is featured on the Discovery Ed homepage after you log in.
Sometimes what's in the news is not always appropriate for the age of students you are working with.
Discovery's Global Wrap is organized so that you can show segments or the entire Wrap.
Ideas for Classroom Use
- Have a Current Events Day of the week.
- Assign a segment to different groups through the Assignment Builder and have them share out.
- Build a writing prompt about different segments to encourage deeper thinking and responses about the media
- Have students select one topic/segment and then have them report out about that particular topic<|endoftext|>
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# Excercise 4.3 Practical Geometry- NCERT Solutions Class 8
Go back to 'Practical Geometry'
## Chapter 4 Ex.4.3 Question 1
Construct the following quadrilaterals.
(i) Quadrilateral $$MORE$$
$$MO = 6\,\rm{cm}$$
$$OR = 4.5\,\rm{cm}$$
$$\angle {M = 60 }^\circ$$
$$\angle {O = 105 }^\circ$$
$$\angle R = 105^\circ$$
(ii) Quadrilateral $$PLAN$$
$$PL=4\,\rm{cm}$$
$$LA=6.5\,\rm{cm}$$
$$\angle P=90^\circ$$
$$\angle A=110^\circ$$
$$\angle N=85^\circ$$
(iii) Parallelogram $$HEAR$$
$$HE = 5\,\rm{cm}$$
$$EA = 6\,\rm{cm}$$
$$\angle R = 85^\circ$$
(iv) Rectangle $$OKAY$$
$$OK=7\,\rm{cm}$$
$$KA=5\,\rm{cm}$$
### Solution
What is known?
Measurements of two sides and three angles
What is unknown?
Construction of a Quadrilateral
Reasoning:
As you are aware, we need five measurements to draw a quadrilateral .
The measurements of two adjacent sides and three angles
Steps:
Let us draw a rough diagram of the quadrilateral
Let us construct the quadrilateral
Step 1: Draw a line segment $$MO=6 \,\rm{cm}.$$
Step 2: With $$M$$ as center draw angle of measure $$60^\circ.$$
Step 3: Draw $$105^\circ$$ from $$O.$$
Step 4: With $$O$$ as center and radius $$4.5\,\rm{cm}$$ draw an arc cutting the ray from $$O$$ at $$R.$$ $$OR = 4.5\,\rm{cm}$$
Step 5: Construct an angle of $$105^{\circ}$$ from $$R$$ as above. The ray from $$R$$ meets the ray from $$M$$ at a point. Mark the intersection point as $$E.$$
Step 6: $$MORE$$ is the required quadrilateral.
Related Sections
Related Sections
Instant doubt clearing with Cuemath Advanced Math Program<|endoftext|>
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# How do you rationalize the denominator and simplify ((sqrt7) - (sqrt 2))/((sqrt7) + (sqrt 2))?
Apr 29, 2016
$\frac{9 - 2 \sqrt{14}}{5}$
#### Explanation:
Given:$\text{ } \textcolor{b r o w n}{\frac{\sqrt{7} - \sqrt{2}}{\sqrt{7} + \sqrt{2}}}$
Trick is to use the scenario:$\text{ } {a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)$
Multiply by 1 but in the form $\textcolor{b l u e}{1 = \frac{\sqrt{7} - \sqrt{2}}{\sqrt{7} - \sqrt{2}}}$
$\textcolor{b r o w n}{\frac{\sqrt{7} - \sqrt{2}}{\sqrt{7} + \sqrt{2}}} \textcolor{b l u e}{\times \frac{\sqrt{7} - \sqrt{2}}{\sqrt{7} - \sqrt{2}}}$
${\left(\sqrt{7} - \sqrt{2}\right)}^{2} / \left({\left(\sqrt{7}\right)}^{2} - {\left(\sqrt{2}\right)}^{2}\right)$
$\frac{7 - 2 \sqrt{7} \sqrt{2} + 2}{7 - 2}$
$\frac{9 - 2 \sqrt{14}}{5}$<|endoftext|>
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# Difference between revisions of "2018 AIME I Problems/Problem 8"
## Problem
Let $ABCDEF$ be an equiangular hexagon such that $AB=6, BC=8, CD=10$, and $DE=12$. Denote by $d$ the diameter of the largest circle that fits inside the hexagon. Find $d^2$.
## Solution 1
First of all, draw a good diagram! This is always the key to solving any geometry problem. Once you draw it, realize that $EF=2, FA=16$. Why? Because since the hexagon is equiangular, we can put an equilateral triangle around it, with side length $6+8+10=24$. Then, if you drew it to scale, notice that the "widest" this circle can be according to $AF, CD$ is $7\sqrt{3}$. And it will be obvious that the sides won't be inside the circle, so our answer is $\boxed{147}$.
-expiLnCalc
## Solution 2
Like solution 1, draw out the large equilateral triangle with side length $24$. Let the tangent point of the circle at $\overline{CD}$ be G and the tangent point of the circle at $\overline{AF}$ be H. Clearly, GH is the diameter of our circle, and is also perpendicular to $\overline{CD}$ and $\overline{AF}$.
The equilateral triangle of side length $10$ is similar to our large equilateral triangle of $24$. And the height of the former equilateral triangle is $\sqrt{10^2-5^2}=5\sqrt{3}$. By our similarity condition, $\frac{10}{24}=\frac{5\sqrt{3}}{d+5\sqrt{3}}$
Solving this equation gives $d=7\sqrt{3}$, and $d^2=\boxed{147}$
~novus677<|endoftext|>
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Four different-sized beakers are filled with water up to the same height. The temperature of the water is same in all four beakers. If we put 100 gram of ice
in each beaker, what effect will occur?
A The temperature of water in beaker 2 will change the most.
B The ice will melt fastest in the beaker 3.
C The temperature changes will be same in all four beakers.
D The temperature of the water in beaker 3 will change the most.
Refer to the given puzzle.
Four components of food are hidden in the given puzzle. Solve the puzzle and select the option which is not hidden in the puzzle.
AIt builds our body and helps in growth.
BIt is an energy giving nutrient that gives more energy than carbohydrate.
CIt helps the body to get rid of undigested food.
DIt keeps our body fit and helps to resist diseases.
how does carbohydrates help in getting rid of undigested food
Which of the following statements are correct?
(i) The glass sheet in a solar cooker is responsible for green house effect.
(ii) Acid rain happens because earth’s atmosphere contains acids.
(iii) Biomass is a renewable source of energy.
(iv) Non-conventional sources of energy are nuclear, solar, biomass etc.
(v) Sun can be taken as inexhaustible source of energy.
A(i) and (iv) only
B (ii) and (iii) only
C (iii), (iv) and (v) only
D (i), (iii) and (v) only
the ans is not correct... (iv) is also correct
Refer to the given flow chart and select the correct option regarding P, Q, R, S and T.
Labeled one of the columns wrong
Read the given statements carefully and select the correct option.
Statement 1 : When an ant bites, it injects formic acid into the skin.
Statement 2 : The effect of the acid can be neutralised by rubbing moist baking soda.
ABoth statements 1 and 2 are true and statement 2 is the correct explanation of statement 1.
BBoth statements 1 and 2 are true but statement 2 is not the correct explanation of statement 1.
CStatement 1 is true but statement 2 is false.
DBoth statements 1 and 2 are false.
The given figure shows the distillation of an aqueous solution of sodium chloride :
What is the temperature shown by the thermometer and what is left in the distillation flask?
A< 100 ------------------ White residue
B85 ---------------------- No residue
C100 --------------------- No residue
D> 100 ------------------ White residue
but the water will start boiling at 100degree
Refer the given figure showing changes in a flag’s shadow during the day.
Shadows P, Q and R represent different times of the day. At what time, would Preeti most likely to have (i) Breakfast and (ii) Evening snacks, respectively?
A(i) - R, (ii) - Q
B(i) - P, (ii) - Q
C(i) - R, (ii) - P
D(i) - P, (ii) - R
Vinod made use of the given flowchart to find out whether a steel rod (object) works like a magnet, or is magnetic or non-magnetic material. The points J, E, F, G and H are called 'exit points'. Which 'exit point' would Vinod get for the steel rod?<|endoftext|>
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3D Printing Furthers Studies into Dental Evolution as Australian Researchers Study Ancient Fish Fossil
If you are going to take a cool, relaxing dip in the ocean, it’s really best just to take your chances and not consider the many other species that are swimming and swirling around in the water beside and beneath you. While you might be worrying about stepping on a stingray, getting stung by a jellyfish, or chomped on by Jaws, there are so many other things you’ve never even considered—and that’s probably good. But while the terrors of the deep may serve as reason to send you running out of the surf, they can also take on a much more benevolent—and educational—form. Especially when we go back a few years—or say, maybe 400 million of them.
Some lucky scientists have been studying an ancient fish fossil found in Australia which is actually helping us to learn more about the evolution of the human body as well. The current subject of fascination at the Australian National University (ANU) and Queensland Museum is teeth. And the secret to how they progressed in human form may be found in the fossil of the Buchanosteus, a placoderm that is certainly long extinct.
“We have used CT scanning facilities to investigate the internal structure of very fragile fossil skulls and braincases that have been acid-etched from limestone rock,” said Gavin Young from ANU.
This allowed them to see how the jaw moved, as well as investigating the internal tissue showing ‘tooth-like denticles.’ The goal was to establish how teeth evolved in animals, and humans. And this has been a major scientific question, explains Yuzhi Hu, a PhD candidate at ANU.
“We are researching this question using new evidence from an exceptionally preserved fossil fish about 400 million years old,” said Hu.
Due to 3D printing, scientists now have a much better way to study a specific and detailed structure such as the jaw and teeth of such a creature. The team discovered that the Buchanosteus jaw and dental pattern is unlike anything alive today, and they were also able to show evidence which disputes previous claims that these ancient fish actually had real teeth.
“Placoderms have been a common focus in the question of tooth origins,” said Carole Burrow from Queensland Museum. “Our team has been able to examine the gnathal plates of placoderms from the Early Devonian period, and compare their internal and external structure with those of younger placoderms as well as with the true teeth in other jawed fishes.”
The researchers outlined their findings in ‘Placoderms and the evolutionary origin of teeth: a comment on Rücklin & Donoghue,’ disputing results of previous research which suggested that the extinct placoderms had real teeth.
“A key question is whether these gnathal plates were modified from external dermal bones, or had ‘denticles’ representing true teeth with pulp cavities. The recent contribution by Rücklin & Donoghue confuses this issue, because their claimed ‘anterior supragnathal’ (ASG) of the placoderm Romundina stellina shows no evidence that it came from the oral cavity, and is more likely an external dermal element,” state the researchers in their paper. “Also, the tissue identified as enameloid is not birefringent and thus not enameloid. Their inferences about growth of toothplates, phylogenetic loss of enameloid, and independent development of teeth and jaws, based on the structure of this plate, are therefore invalid.”
Through their digital study as well as being able to study 3D models, the researchers stated that they found no ‘orthodentine,’ as would be typical in a tooth. With this, they concluded that there indeed were not teeth, but only the tubercles. There is also a reply to this paper, which you can read here, as the scientists bat the subject of what really constitutes ancient fish teeth back and forth.
And despite the outcome of the research—or how the ongoing argument regarding fossilized dental patterns ensues—one thing is for certain here: we are seeing true evidence of the benefits provided to science by 3D printing, as the scientists were allowed to re-create fossils affordably and study them comprehensively, while saving the originals from further degradation. This is a growing trend today in both labs and museums, which should lend many other great successes to research as scientists continue to share—and discuss subjects such as these ancient fossils at hand. What do you think of this study? Let’s talk about it over in the 3D Printed Fish Fossil forum at 3DPB.com.[Source: Deccan Chronicle]
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Indian religions, sometimes also termed as Dharmic religions, are the religions that originated in the Indian subcontinent; namely Hinduism, Jainism, Buddhism, and Sikhism.[web 1][note 1] These religions are also all classified as Eastern religions. Although Indian religions are connected through the history of India, they constitute a wide range of religious communities, and are not confined to the Indian subcontinent.[web 1]
Evidence attesting to prehistoric religion in the Indian subcontinent derives from scattered Mesolithic rock paintings. The Harappan people of the Indus Valley Civilisation, which lasted from 3300 to 1300 BCE (mature period 2600–1900 BCE), had an early urbanized culture which predates the Vedic religion.
The documented history of Indian religions begins with the historical Vedic religion, the religious practices of the early Indo-Iranians, which were collected and later redacted into the Vedas. The period of the composition, redaction and commentary of these texts is known as the Vedic period, which lasted from roughly 1750 to 500 BCE. The philosophical portions of the Vedas were summarized in Upanishads, which are commonly referred to as Vedānta, variously interpreted to mean either the "last chapters, parts of the Veda" or "the object, the highest purpose of the Veda". The early Upanishads all predate the Common Era, five[note 2] of the eleven principal Upanishads were composed in all likelihood before 6th century BCE, and contain the earliest mentions of Yoga and Moksha.
The Reform or Shramanic Period between 800 and 200 BCE marks a "turning point between the Vedic Hinduism and Puranic Hinduism". The Shramana movement, an ancient Indian religious movement parallel to but separate from Vedic tradition, often defied many of the Vedic and Upanishadic concepts of soul (Atman) and the ultimate reality (Brahman). In 6th century BCE, the Shramnic movement matured into Jainism and Buddhism and was responsible for the schism of Indian religions into two main philosophical branches of astika, which venerates Veda (e.g., six orthodox schools of Hinduism) and nastika (e.g., Buddhism, Jainism, Charvaka, etc.). However, both branches shared the related concepts of Yoga, saṃsāra (the cycle of birth and death) and moksha (liberation from that cycle).[note 3][note 4]
The Puranic Period (200 BCE – 500 CE) and Early Medieval period (500–1100 CE) gave rise to new configurations of Hinduism, especially bhakti and Shaivism, Shaktism, Vaishnavism, Smarta and much smaller groups like the conservative Shrauta.
The early Islamic period (1100–1500 CE) also gave rise to new movements. Sikhism was founded in the 15th century on the teachings of Guru Nanak and the nine successive Sikh Gurus in Northern India.[web 2] The vast majority of its adherents originate in the Punjab region.
James Mill (1773–1836), in his The History of British India (1817), distinguished three phases in the history of India, namely Hindu, Muslim and British civilisations. This periodisation has been criticised, for the misconceptions it has given rise to. Another periodisation is the division into "ancient, classical, medieval and modern periods", although this periodization has also received criticism.
Romila Thapar notes that the division of Hindu-Muslim-British periods of Indian history gives too much weight to "ruling dynasties and foreign invasions," neglecting the social-economic history which often showed a strong continuity. The division in Ancient-Medieval-Modern overlooks the fact that the Muslim-conquests took place between the eight and the fourteenth century, while the south was never completely conquered. According to Thapar, a periodisation could also be based on "significant social and economic changes," which are not strictly related to a change of ruling powers.[note 5]
Smart and Michaels seem to follow Mill's periodisation, while Flood and Muesse follow the "ancient, classical, mediaeval and modern periods" periodisation. An elaborate periodisation may be as follows:
Evidence attesting to prehistoric religion in the Indian subcontinent derives from scattered Mesolithic rock paintings such as at Bhimbetka, depicting dances and rituals. Neolithic agriculturalists inhabiting the Indus River Valley buried their dead in a manner suggestive of spiritual practices that incorporated notions of an afterlife and belief in magic. Other South Asian Stone Age sites, such as the Bhimbetka rock shelters in central Madhya Pradesh and the Kupgal petroglyphs of eastern Karnataka, contain rock art portraying religious rites and evidence of possible ritualised music.[web 3]
The religion and belief system of the Indus valley people have received considerable attention, especially from the view of identifying precursors to deities and religious practices of Indian religions that later developed in the area. However, due to the sparsity of evidence, which is open to varying interpretations, and the fact that the Indus script remains undeciphered, the conclusions are partly speculative and largely based on a retrospective view from a much later Hindu perspective. An early and influential work in the area that set the trend for Hindu interpretations of archaeological evidence from the Harrapan sites was that of John Marshall, who in 1931 identified the following as prominent features of the Indus religion: a Great Male God and a Mother Goddess; deification or veneration of animals and plants; symbolic representation of the phallus (linga) and vulva (yoni); and, use of baths and water in religious practice. Marshall's interpretations have been much debated, and sometimes disputed over the following decades.
One Indus valley seal shows a seated, possibly ithyphallic and tricephalic, figure with a horned headdress, surrounded by animals. Marshall identified the figure as an early form of the Hindu god Shiva (or Rudra), who is associated with asceticism, yoga, and linga; regarded as a lord of animals; and often depicted as having three eyes. The seal has hence come to be known as the Pashupati Seal, after Pashupati (lord of all animals), an epithet of Shiva. While Marshall's work has earned some support, many critics and even supporters have raised several objections. Doris Srinivasan has argued that the figure does not have three faces, or yogic posture, and that in Vedic literature Rudra was not a protector of wild animals. Herbert Sullivan and Alf Hiltebeitel also rejected Marshall's conclusions, with the former claiming that the figure was female, while the latter associated the figure with Mahisha, the Buffalo God and the surrounding animals with vahanas (vehicles) of deities for the four cardinal directions. Writing in 2002, Gregory L. Possehl concluded that while it would be appropriate to recognise the figure as a deity, its association with the water buffalo, and its posture as one of ritual discipline, regarding it as a proto-Shiva would be going too far. Despite the criticisms of Marshall's association of the seal with a proto-Shiva icon, it has been interpreted as the Tirthankara Rishabha by Jains and Dr. Vilas Sangave or an early Buddha by Buddhists. Historians like Heinrich Zimmer, Thomas McEvilley are of the opinion that there exists some link between first Jain Tirthankara Rishabha and Indus Valley civilisation.
Marshall hypothesized the existence of a cult of Mother Goddess worship based upon excavation of several female figurines, and thought that this was a precursor of the Hindu sect of Shaktism. However the function of the female figurines in the life of Indus Valley people remains unclear, and Possehl does not regard the evidence for Marshall's hypothesis to be "terribly robust". Some of the baetyls interpreted by Marshall to be sacred phallic representations are now thought to have been used as pestles or game counters instead, while the ring stones that were thought to symbolise yoni were determined to be architectural features used to stand pillars, although the possibility of their religious symbolism cannot be eliminated. Many Indus Valley seals show animals, with some depicting them being carried in processions, while others show chimeric creations. One seal from Mohen-jodaro shows a half-human, half-buffalo monster attacking a tiger, which may be a reference to the Sumerian myth of such a monster created by goddess Aruru to fight Gilgamesh.
In contrast to contemporary Egyptian and Mesopotamian civilisations, Indus valley lacks any monumental palaces, even though excavated cities indicate that the society possessed the requisite engineering knowledge. This may suggest that religious ceremonies, if any, may have been largely confined to individual homes, small temples, or the open air. Several sites have been proposed by Marshall and later scholars as possibly devoted to religious purpose, but at present only the Great Bath at Mohenjo-daro is widely thought to have been so used, as a place for ritual purification. The funerary practices of the Harappan civilisation is marked by its diversity with evidence of supine burial; fractional burial in which the body is reduced to skeletal remains by exposure to the elements before final interment; and even cremation.
The early Dravidian religion constituted of non-Vedic form of Hinduism in that they were either historically or are at present Āgamic. The Agamas are non-vedic in origin and have been dated either as post-vedic texts. or as pre-vedic oral compositions. The Agamas are a collection of Tamil and later Sanskrit scriptures chiefly constituting the methods of temple construction and creation of murti, worship means of deities, philosophical doctrines, meditative practices, attainment of sixfold desires and four kinds of yoga. The worship of tutelary deity, sacred flora and fauna in Hinduism is also recognized as a survival of the pre-Vedic Dravidian religion.
Ancient Tamil grammatical works Tolkappiyam, the ten anthologies Pattuppāṭṭu, the eight anthologies Eṭṭuttokai also sheds light on early religion of ancient Dravidians. Seyon was glorified as the red god seated on the blue peacock, who is ever young and resplendent, as the favored god of the Tamils. Sivan was also seen as the supreme God. Early iconography of Seyyon and Sivan and their association with native flora and fauna goes back to Indus Valley Civilization. The Sangam landscape was classified into five categories, thinais, based on the mood, the season and the land. Tolkappiyam, mentions that each of these thinai had an associated deity such Seyyon in Kurinji-the hills, Thirumaal in Mullai-the forests, and Kotravai in Marutham-the plains, and Wanji-ko in the Neithal-the coasts and the seas. Other gods mentioned were Mayyon and Vaali who were all assimilated into Hinduism over time. Dravidian linguistic influence on early Vedic religion is evident, many of these features are already present in the oldest known Indo-Aryan language, the language of the Rigveda (c. 1500 BCE), which also includes over a dozen words borrowed from Dravidian. This represents an early religious and cultural fusion[note 7] or synthesis between ancient Dravidians and Indo-Aryans, which became more evident over time with sacred iconography, traditions, philosophy, flora and fauna that went on to influence Hinduism, Buddhism, Charvaka, Sramana and Jainism.
Throughout Tamilakam, a king was considered to be divine by nature and possessed religious significance. The king was 'the representative of God on earth’ and lived in a “koyil”, which means the “residence of a god”. The Modern Tamil word for temple is koil. Titual worship was also given to kings. Modern words for god like “kō” (“king”), “iṟai” (“emperor”) and “āṇḍavar” ( “conqueror”) now primarily refer to gods. These elements were incorporated later into Hinduism like the legendary marriage of Shiva to Queen Mīnātchi who ruled Madurai or Wanji-ko, a god who later merged into Indra. Tolkappiyar refers to the Three Crowned Kings as the “Three Glorified by Heaven”. In the Dravidian-speaking South, the concept of divine kingship led to the assumption of major roles by state and temple.
The cult of the mother goddess is treated as an indication of a society which venerated femininity. This mother goddess was conceived as a virgin, one who has given birth to all and one, typically associated with Shaktism. The temples of the Sangam days, mainly of Madurai, seem to have had priestesses to the deity, which also appear predominantly a goddess. In the Sangam literature, there is an elaborate description of the rites performed by the Kurava priestess in the shrine Palamutircholai. Among the early Dravidians the practice of erecting memorial stones “Natukal or Hero Stone had appeared, and it continued for quite a long time after the Sangam age, down to about 16th century. It was customary for people who sought victory in war to worship these hero stones to bless them with victory.
The documented history of Indian religions begins with the historical Vedic religion, the religious practices of the early Indo-Aryans, which were collected and later redacted into the Samhitas (usually known as the Vedas), four canonical collections of hymns or mantras composed in archaic Sanskrit. These texts are the central shruti (revealed) texts of Hinduism. The period of the composition, redaction and commentary of these texts is known as the Vedic period, which lasted from roughly 1750 to 500 BCE.
The Vedic Period is most significant for the composition of the four Vedas, Brahmanas and the older Upanishads (both presented as discussions on the rituals, mantras and concepts found in the four Vedas), which today are some of the most important canonical texts of Hinduism, and are the codification of much of what developed into the core beliefs of Hinduism.
Some modern Hindu scholars use the "Vedic religion" synonymously with "Hinduism." According to Sundararajan, Hinduism is also known as the Vedic religion. Other authors state that the Vedas contain "the fundamental truths about Hindu Dharma" which is called "the modern version of the ancient Vedic Dharma" The Arya Samajis recognize the Vedic religion as true Hinduism. Nevertheless, according to Jamison and Witzel,
... to call this period Vedic Hinduism is a contradiction in terms since Vedic religion is very different from what we generally call Hindu religion – at least as much as Old Hebrew religion is from medieval and modern Christian religion. However, Vedic religion is treatable as a predecessor of Hinduism."[note 8]
The mode of worship was the performance of Yajna, sacrifices which involved sacrifice and sublimation of the havana sámagri (herbal preparations) in the fire, accompanied by the singing of Samans and 'mumbling' of Yajus, the sacrificial mantras. The sublime meaning of the word yajna is derived from the Sanskrit verb yaj, which has a three-fold meaning of worship of deities (devapujana), unity (saògatikaraña) and charity (dána). An essential element was the sacrificial fire – the divine Agni – into which oblations were poured, as everything offered into the fire was believed to reach God.
Central concepts in the Vedas are Satya and Rta. Satya is derived from Sat, the present participle of the verbal root as, "to be, to exist, to live". Sat means "that which really exists [...] the really existent truth; the Good", and Sat-ya means "is-ness". Rta, "that which is properly joined; order, rule; truth", is the principle of natural order which regulates and coordinates the operation of the universe and everything within it. "Satya (truth as being) and rita (truth as law) are the primary principles of Reality and its manifestation is the background of the canons of dharma, or a life of righteousness." "Satya is the principle of integration rooted in the Absolute, rita is its application and function as the rule and order operating in the universe." Conformity with Ṛta would enable progress whereas its violation would lead to punishment. Panikkar remarks:
Ṛta is the ultimate foundation of everything; it is "the supreme", although this is not to be understood in a static sense. [...] It is the expression of the primordial dynamism that is inherent in everything...."
The term rta is inherited from the Proto-Indo-Iranian religion, the religion of the Indo-Iranian peoples prior to the earliest Vedic (Indo-Aryan) and Zoroastrian (Iranian) scriptures. "Asha" is the Avestan language term (corresponding to Vedic language ṛta) for a concept of cardinal importance to Zoroastrian theology and doctrine. The term "dharma" was already used in Brahmanical thought, where it was conceived as an aspect of Rta.
The Vedic religion evolved into Hinduism and Vedanta, a religious path considering itself the 'essence' of the Vedas, interpreting the Vedic pantheon as a unitary view of the universe with 'God' (Brahman) seen as immanent and transcendent in the forms of Ishvara and Brahman. This post-Vedic systems of thought, along with the Upanishads and later texts like epics (namely Gita of Mahabharat), is a major component of modern Hinduism. The ritualistic traditions of Vedic religion are preserved in the conservative Śrauta tradition.
Since Vedic times, "people from many strata of society throughout the subcontinent tended to adapt their religious and social life to Brahmanic norms", a process sometimes called Sanskritization. It is reflected in the tendency to identify local deities with the gods of the Sanskrit texts.
During the time of the shramanic reform movements "many elements of the Vedic religion were lost". According to Michaels, "it is justified to see a turning point between the Vedic religion and Hindu religions".
The late Vedic period (9th to 6th centuries BCE) marks the beginning of the Upanisadic or Vedantic period.[web 4][note 10][note 11] This period heralded the beginning of much of what became classical Hinduism, with the composition of the Upanishads,:183 later the Sanskrit epics, still later followed by the Puranas.
Upanishads form the speculative-philosophical basis of classical Hinduism and are known as Vedanta (conclusion of the Vedas). The older Upanishads launched attacks of increasing intensity on the ritual. Anyone who worships a divinity other than the Self is called a domestic animal of the gods in the Brihadaranyaka Upanishad. The Mundaka launches the most scathing attack on the ritual by comparing those who value sacrifice with an unsafe boat that is endlessly overtaken by old age and death.
Jainism and Buddhism belong to the sramana tradition. These religions rose into prominence in 700–500 BCE in the Magadha kingdom., reflecting "the cosmology and anthropology of a much older, pre-Aryan upper class of northeastern India", and were responsible for the related concepts of saṃsāra (the cycle of birth and death) and moksha (liberation from that cycle).[note 12]
The shramana movements challenged the orthodoxy of the rituals. The shramanas were wandering ascetics distinct from Vedism.[note 13][note 14][note 15] Mahavira, proponent of Jainism, and Buddha (c. 563-483), founder of Buddhism were the most prominent icons of this movement.
Shramana gave rise to the concept of the cycle of birth and death, the concept of samsara, and the concept of liberation.[note 16][note 17][note 18][note 19] The influence of Upanishads on Buddhism has been a subject of debate among scholars. While Radhakrishnan, Oldenberg and Neumann were convinced of Upanishadic influence on the Buddhist canon, Eliot and Thomas highlighted the points where Buddhism was opposed to Upanishads. Buddhism may have been influenced by some Upanishadic ideas, it however discarded their orthodox tendencies. In Buddhist texts Buddha is presented as rejecting avenues of salvation as "pernicious views".
The 24th Tirthankara of Jainism, Mahavira, stressed five vows, including ahimsa (non-violence), satya (truthfulness), asteya (non-stealing) and aparigraha (non-attachment). Jain orthodoxy believes the teachings of the Tirthankaras predates all known time and scholars believe Parshva, accorded status as the 23rd Tirthankara, was a historical figure. The Vedas are believed to have documented a few Tirthankaras and an ascetic order similar to the shramana movement.[note 21]
Buddhism was historically founded by Siddhartha Gautama, a Kshatriya prince-turned-ascetic, and was spread beyond India through missionaries. It later experienced a decline in India, but survived in Nepal and Sri Lanka, and remains more widespread in Southeast and East Asia.
Gautama Buddha, who was called an "awakened one" (Buddha), was born into the Shakya clan living at Kapilavastu and Lumbini in what is now southern Nepal. The Buddha was born at Lumbini, as emperor Ashoka's Lumbini pillar records, just before the kingdom of Magadha (which traditionally is said to have lasted from c. 546–324 BCE) rose to power. The Shakyas claimed Angirasa and Gautama Maharishi lineage, via descent from the royal lineage of Ayodhya.
Buddhism emphasises enlightenment (nibbana, nirvana) and liberation from the rounds of rebirth. This objective is pursued through two schools, Theravada, the Way of the Elders (practised in Sri Lanka, Burma, Thailand, SE Asia, etc.) and Mahayana, the Greater Way (practised in Tibet, China, Japan etc.). There may be some differences in the practice between the two schools in reaching the objective. In the Theravada practice this is pursued in seven stages of purification (visuddhi); viz. physical purification by taking precepts (sila visiddhi), mental purification by insight meditation (citta visuddhi), followed by purification of views and concepts (ditthi visuddhi), purification by overcoming of doubts (kinkha vitarana vishuddhi), purification by acquiring knowledge and wisdom of the right path (maggarmagga-nanadasana visuddhi), attaining knowledge and wisdom through the course of practice (patipada-nanadasana visuddhi), and purification by attaining knowledge and insight wisdom (nanadasana visuddhi).
Both Jainism and Buddhism spread throughout India during the period of the Magadha empire.
Buddhism in India spread during the reign of Ashoka of the Maurya Empire, who patronised Buddhist teachings and unified the Indian subcontinent in the 3rd century BCE. He sent missionaries abroad, allowing Buddhism to spread across Asia. Jainism began its golden period during the reign of Emperor Kharavela of Kalinga in the 2nd century BCE.
Flood and Muesse take the period between 200 BCE and 500 BCE as a separate period, in which the epics and the first puranas were being written. Michaels takes a greater timespan, namely the period between 200 BCE and 1100 CE, which saw the rise of so-called "Classical Hinduism", with its "golden age" during the Gupta Empire.
According to Alf Hiltebeitel, a period of consolidation in the development of Hinduism took place between the time of the late Vedic Upanishad (c. 500 BCE) and the period of the rise of the Guptas (c. 320–467 CE), which he calls the "Hindus synthesis", "Brahmanic synthesis", or "orthodox synthesis". It develops in interaction with other religions and peoples:
The emerging self-definitions of Hinduism were forged in the context of continuous interaction with heterodox religions (Buddhists, Jains, Ajivikas) throughout this whole period, and with foreign people (Yavanas, or Greeks; Sakas, or Scythians; Pahlavas, or Parthians; and Kusanas, or Kushans) from the third phase on [between the Mauryan empire and the rise of the Guptas].
The end of the Vedantic period around the 2nd century CE spawned a number of branches that furthered Vedantic philosophy, and which ended up being seminaries in their own right. Prominent amongst these developers were Yoga, Dvaita, Advaita and the medieval Bhakti movement.
The smriti texts of the period between 200 BCE-100 CE proclaim the authority of the Vedas, and "nonrejection of the Vedas comes to be one of the most important touchstones for defining Hinduism over and against the heterodoxies, which rejected the Vedas." Of the six Hindu darsanas, the Mimamsa and the Vedanta "are rooted primarily in the Vedic sruti tradition and are sometimes called smarta schools in the sense that they develop smarta orthodox current of thoughts that are based, like smriti, directly on sruti." According to Hiltebeitel, "the consolidation of Hinduism takes place under the sign of bhakti." It is the Bhagavadgita that seals this achievement. The result is a universal achievement that may be called smarta. It views Shiva and Vishnu as "complementary in their functions but ontologically identical".
In earlier writings, Sanskrit 'Vedānta' simply referred to the Upanishads, the most speculative and philosophical of the Vedic texts. However, in the medieval period of Hinduism, the word Vedānta came to mean the school of philosophy that interpreted the Upanishads. Traditional Vedānta considers scriptural evidence, or shabda pramāna, as the most authentic means of knowledge, while perception, or pratyaksa, and logical inference, or anumana, are considered to be subordinate (but valid).
The systematisation of Vedantic ideas into one coherent treatise was undertaken by Badarāyana in the Brahma Sutras which was composed around 200 BCE. The cryptic aphorisms of the Brahma Sutras are open to a variety of interpretations. This resulted in the formation of numerous Vedanta schools, each interpreting the texts in its own way and producing its own sub-commentaries.
After 200 CE several schools of thought were formally codified in Indian philosophy, including Samkhya, Yoga, Nyaya, Vaisheshika, Mimāṃsā and Advaita Vedanta. Hinduism, otherwise a highly polytheistic, pantheistic or monotheistic religion, also tolerated atheistic schools. The thoroughly materialistic and anti-religious philosophical Cārvāka school that originated around the 6th century BCE is the most explicitly atheistic school of Indian philosophy. Cārvāka is classified as a nāstika ("heterodox") system; it is not included among the six schools of Hinduism generally regarded as orthodox. It is noteworthy as evidence of a materialistic movement within Hinduism. Our understanding of Cārvāka philosophy is fragmentary, based largely on criticism of the ideas by other schools, and it is no longer a living tradition. Other Indian philosophies generally regarded as atheistic include Samkhya and Mimāṃsā.
Two of Hinduism's most revered epics, the Mahabharata and Ramayana were compositions of this period. Devotion to particular deities was reflected from the composition of texts composed to their worship. For example, the Ganapati Purana was written for devotion to Ganapati (or Ganesh). Popular deities of this era were Shiva, Vishnu, Durga, Surya, Skanda, and Ganesh (including the forms/incarnations of these deities).
In the latter Vedantic period, several texts were also composed as summaries/attachments to the Upanishads. These texts collectively called as Puranas allowed for a divine and mythical interpretation of the world, not unlike the ancient Hellenic or Roman religions. Legends and epics with a multitude of gods and goddesses with human-like characteristics were composed.
The Gupta period marked a watershed of Indian culture: the Guptas performed Vedic sacrifices to legitimize their rule, but they also patronized Buddhism, which continued to provide an alternative to Brahmanical orthodoxy. Buddhism continued to have a significant presence in some regions of India until the 12th century.
Tantrism originated in the early centuries CE and developed into a fully articulated tradition by the end of the Gupta period. According to Michaels this was the "Golden Age of Hinduism" (c. 320–650 CE), which flourished during the Gupta Empire (320 to 550 CE) until the fall of the Harsha Empire (606 to 647 CE). During this period, power was centralised, along with a growth of far distance trade, standardizarion of legal procedures, and general spread of literacy. Mahayana Buddhism flourished, but the orthodox Brahmana culture began to be rejuvenated by the patronage of the Gupta Dynasty. The position of the Brahmans was reinforced, and the first Hindu temples emerged during the late Gupta age.
After the end of the Gupta Empire and the collapse of the Harsha Empire, power became decentralised in India. Several larger kingdoms emerged, with "countless vasal states".[note 23] The kingdoms were ruled via a feudal system. Smaller kingdoms were dependent on the protection of the larger kingdoms. "The great king was remote, was exalted and deified", as reflected in the Tantric Mandala, which could also depict the king as the centre of the mandala.
The disintegration of central power also lead to regionalisation of religiosity, and religious rivalry.[note 24] Local cults and languages were enhanced, and the influence of "Brahmanic ritualistic Hinduism" was diminished. Rural and devotional movements arose, along with Shaivism, Vaisnavism, Bhakti and Tantra, though "sectarian groupings were only at the beginning of their development". Religious movements had to compete for recognition by the local lords. Buddhism lost its position, and began to disappear in India.
In the same period Vedanta changed, incorporating Buddhist thought and its emphasis on consciousness and the working of the mind. Buddhism, which was supported by the ancient Indian urban civilisation lost influence to the traditional religions, which were rooted in the countryside. In Bengal, Buddhism was even prosecuted. But at the same time, Buddhism was incorporated into Hinduism, when Gaudapada used Buddhist philosophy to reinterpret the Upanishads. This also marked a shift from Atman and Brahman as a "living substance" to "maya-vada"[note 25], where Atman and Brahman are seen as "pure knowledge-consciousness". According to Scheepers, it is this "maya-vada" view which has come to dominate Indian thought.
The Bhakti movement began with the emphasis on the worship of God, regardless of one's status – whether priestly or laypeople, men or women, higher social status or lower social status. The movements were mainly centered on the forms of Vishnu (Rama and Krishna) and Shiva. There were however popular devotees of this era of Durga. The best-known devotees are the Nayanars from southern India. The most popular Shaiva teacher of the south was Basava, while of the north it was Gorakhnath. Female saints include figures like Akkamadevi, Lalleshvari and Molla.
The "alwar" or "azhwars" (Tamil: ஆழ்வார்கள், āzvārkaḷ [aːɻʋaːr], those immersed in god) were Tamil poet-saints of south India who lived between the 6th and 9th centuries CE and espoused "emotional devotion" or bhakti to Visnu-Krishna in their songs of longing, ecstasy and service. The most popular Vaishnava teacher of the south was Ramanuja, while of the north it was Ramananda.
Several important icons were women. For example, within the Mahanubhava sect, the women outnumbered the men, and administration was many times composed mainly of women. Mirabai is the most popular female saint in India.
According to The Centre for Cultural Resources and Training,
Vaishanava bhakti literature was an all-India phenomenon, which started in the 6th–7th century A.D. in the Tamil-speaking region of South India, with twelve Alvar (one immersed in God) saint-poets, who wrote devotional songs. The religion of Alvar poets, which included a woman poet, Andal, was devotion to God through love (bhakti), and in the ecstasy of such devotions they sang hundreds of songs which embodied both depth of feeling and felicity of expressions [web 8]
In the 12th and 13th centuries, Turks and Afghans invaded parts of northern India and established the Delhi Sultanate in the former Rajput holdings. The subsequent Slave dynasty of Delhi managed to conquer large areas of northern India, approximately equal in extent to the ancient Gupta Empire, while the Khalji dynasty conquered most of central India but were ultimately unsuccessful in conquering and uniting the subcontinent. The Sultanate ushered in a period of Indian cultural renaissance. The resulting "Indo-Muslim" fusion of cultures left lasting syncretic monuments in architecture, music, literature, religion, and clothing.
During the 14th to 17th centuries, a great Bhakti movement swept through central and northern India, initiated by a loosely associated group of teachers or Sants. Ramananda, Ravidas, Srimanta Sankardeva, Chaitanya Mahaprabhu, Vallabha Acharya, Sur, Meera, Kabir, Tulsidas, Namdev, Dnyaneshwar, Tukaram and other mystics spearheaded the Bhakti movement in the North while Annamacharya, Bhadrachala Ramadas, Tyagaraja among others propagated Bhakti in the South. They taught that people could cast aside the heavy burdens of ritual and caste, and the subtle complexities of philosophy, and simply express their overwhelming love for God. This period was also characterized by a spate of devotional literature in vernacular prose and poetry in the ethnic languages of the various Indian states or provinces.
Lingayatism is a distinct Shaivite tradition in India, established in the 12th century by the philosopher and social reformer Basavanna. The adherents of this tradition are known as Lingayats. The term is derived from Lingavantha in Kannada, meaning 'one who wears Ishtalinga on their body' (Ishtalinga is the representation of the God). In Lingayat theology, Ishtalinga is an oval-shaped emblem symbolising Parasiva, the absolute reality. Contemporary Lingayatism follows a progressive reform–based theology propounded, which has great influence in South India, especially in the state of Karnataka.
According to Nicholson, already between the 12th and 16th century,
... certain thinkers began to treat as a single whole the diverse philosophival teachings of the Upanishads, epics, Puranas, and the schools known retrospectively as the "six systems" (saddarsana) of mainstream Hindu philosophy.
The tendency of "a blurring of philosophical distinctions" has also been noted by Burley. Lorenzen locates the origins of a distinct Hindu identity in the interaction between Muslims and Hindus, and a process of "mutual self-definition with a contrasting Muslim other", which started well before 1800. Both the Indian and the European thinkers who developed the term "Hinduism" in the 19th century were influenced by these philosophers.
Sikhism originated in 15th-century Punjab, Delhi Sultanate (present-day India and Pakistan) with the teachings of Nanak and nine successive gurus. The principal belief in Sikhism is faith in Vāhigurū— represented by the sacred symbol of ēk ōaṅkār [meaning one god]. Sikhism's traditions and teachings are distinctly associated with the history, society and culture of the Punjab. Adherents of Sikhism are known as Sikhs (students or disciples) and number over 27 million across the world.
According to Gavin Flood, the modern period in India begins with the first contacts with western nations around 1500. The period of Mughal rule in India saw the rise of new forms of religiosity.
In the 19th century, under influence of the colonial forces, a synthetic vision of Hinduism was formulated by Raja Ram Mohan Roy, Swami Vivekananda, Sri Aurobindo, Sarvepalli Radhakrishnan and Mahatma Gandhi. These thinkers have tended to take an inclusive view of India's religious history, emphasising the similarities between the various Indian religions.
The modern era has given rise to dozens of Hindu saints with international influence. For example, Brahma Baba established the Brahma Kumaris, one of the largest new Hindu religious movements which teaches the discipline of Raja Yoga to millions. Representing traditional Gaudiya Vaishnavism, Prabhupada founded the Hare Krishna movement, another organisation with a global reach. In late 18th-century India, Swaminarayan founded the Swaminarayan Sampraday. Anandamurti, founder of the Ananda Marga, has also influenced many worldwide. Through the international influence of all of these new Hindu denominations, many Hindu practices such as yoga, meditation, mantra, divination, and vegetarianism have been adopted by new converts.
Jainism continues to be an influential religion and Jain communities live in Indian states Gujarat, Rajasthan, Madhya Pradesh, Maharashtra, Karnataka and Tamil Nadu. Jains authored several classical books in different Indian languages for a considerable period of time.
The Dalit Buddhist movement also referred to as Navayana is a 19th- and 20th-century Buddhist revival movement in India. It received its most substantial impetus from B. R. Ambedkar's call for the conversion of Dalits to Buddhism in 1956 and the opportunity to escape the caste-based society that considered them to be the lowest in the hierarchy.
According to Tilak, the religions of India can be interpreted "differentially" or "integrally", that is by either highlighting the differences or the similarities. According to Sherma and Sarma, western Indologists have tended to emphasise the differences, while Indian Indologists have tended to emphasise the similarities.
Hinduism, Buddhism, Jainism, and Sikhism share certain key concepts, which are interpreted differently by different groups and individuals. Until the 19th century, adherents of those various religions did not tend to label themselves as in opposition to each other, but "perceived themselves as belonging to the same extended cultural family."
Common traits can also be observed in ritual. The head-anointing ritual of abhiseka is of importance in three of these distinct traditions, excluding Sikhism (in Buddhism it is found within Vajrayana). Other noteworthy rituals are the cremation of the dead, the wearing of vermilion on the head by married women, and various marital rituals. In literature, many classical narratives and purana have Hindu, Buddhist or Jain versions.[web 9] All four traditions have notions of karma, dharma, samsara, moksha and various forms of Yoga.
Rama is a heroic figure in all of these religions. In Hinduism he is the God-incarnate in the form of a princely king; in Buddhism, he is a Bodhisattva-incarnate; in Jainism, he is the perfect human being. Among the Buddhist Ramayanas are: Vessantarajataka, Reamker, Ramakien, Phra Lak Phra Lam, Hikayat Seri Rama etc. There also exists the Khamti Ramayana among the Khamti tribe of Asom wherein Rama is an Avatar of a Bodhisattva who incarnates to punish the demon king Ravana (B.Datta 1993). The Tai Ramayana is another book retelling the divine story in Asom.
For a Hindu, dharma is his duty. For a Jain, dharma is righteousness, his conduct. For a Buddhist, dharma is usually taken to be the Buddha's teachings.
Indian mythology also reflects the competition between the various Indian religions. A popular story tells how Vajrapani kills Mahesvara, a manifestation of Shiva depicted as an evil being. The story occurs in several scriptures, most notably the Sarvatathagatatattvasamgraha and the Vajrapany-abhiseka-mahatantra.[note 26] According to Kalupahana, the story "echoes" the story of the conversion of Ambattha. It is to be understood in the context of the competition between Buddhist institutions and Shaivism.
Āstika and nāstika are variously defined terms sometimes used to categorise Indian religions. The traditional definition, followed by Adi Shankara, classifies religions and persons as āstika and nāstika according to whether they accept the authority of the main Hindu texts, the Vedas, as supreme revealed scriptures, or not. By this definition, Nyaya, Vaisheshika, Samkhya, Yoga, Purva Mimamsa and Vedanta are classified as āstika schools, while Charvaka is classified as a nāstika school. Buddhism and Jainism are also thus classified as nāstika religions since they do not accept the authority of the Vedas.
Another set of definitions—notably distinct from the usage of Hindu philosophy—loosely characterise āstika as "theist" and nāstika as "atheist". By these definitions, Sāṃkhya can be considered a nāstika philosophy, though it is traditionally classed among the Vedic āstika schools. From this point of view, Buddhism and Jainism remain nāstika religions.
Buddhists and Jains have disagreed that they are nastika and have redefined the phrases āstika and nāstika in their own view. Jains assign the term nastika to one who is ignorant of the meaning of the religious texts, or those who deny the existence of the soul was well known to the Jainas.
Frawley and Malhotra use the term "Dharmic traditions" to highlight the similarities between the various Indian religions.[note 27] According to Frawley, "all religions in India have been called the Dharma", and can be
...put under the greater umbrella of "Dharmic traditions" which we can see as Hinduism or the spiritual traditions of India in the broadest sense.
According to Paul Hacker, as described by Halbfass, the term "dharma"
...assumed a fundamentally new meaning and function in modern Indian thought, beginning with Bankim Chandra Chatterjee in the nineteenth century. This process, in which dharma was presented as an equivalent of, but also a response to, the western notion of "religion", reflects a fundamental change in the Hindu sense of identity and in the attitude toward other religious and cultural traditions. The foreign tools of "religion" and "nation" became tools of self-definition, and a new and precarious sense of the "unity of Hinduism" and of national as well as religious identity took root.
The emphasis on the similarities and integral unity of the dharmic faiths has been criticised for neglecting the vast differences between and even within the various Indian religions and traditions. According to Richard E. King it is typical of the "inclusivist appropriation of other traditions" of Neo-Vedanta:
The inclusivist appropriation of other traditions, so characteristic of neo-Vedanta ideology, appears on three basic levels. First, it is apparent in the suggestion that the (Advaita) Vedanta philosophy of Sankara (c. eighth century CE) constitutes the central philosophy of Hinduism. Second, in an Indian context, neo-Vedanta philosophy subsumes Buddhist philosophies in terms of its own Vedantic ideology. The Buddha becomes a member of the Vedanta tradition, merely attempting to reform it from within. Finally, at a global level, neo-Vedanta colonizes the religious traditions of the world by arguing for the centrality of a non-dualistic position as the philosophia perennis underlying all cultural differences.
The inclusion of Buddhists, Jains and Sikhs within Hinduism is part of the Indian legal system. The 1955 Hindu Marriage Act "[defines] as Hindus all Buddhists, Jains, Sikhs and anyone who is not a Christian, Muslim, Parsee (Zoroastrian) or Jew". And the Indian Constitution says that "reference to Hindus shall be construed as including a reference to persons professing the Sikh, Jaina or Buddhist religion".
In a judicial reminder, the Indian Supreme Court observed Sikhism and Jainism to be sub-sects or special faiths within the larger Hindu fold,[web 10][note 28] and that Jainism is a denomination within the Hindu fold.[web 10][note 29] Although the government of British India counted Jains in India as a major religious community right from the first Census conducted in 1873, after independence in 1947 Sikhs and Jains were not treated as national minorities.[web 10][note 30] In 2005 the Supreme Court of India declined to issue a writ of Mandamus granting Jains the status of a religious minority throughout India. The Court however left it to the respective states to decide on the minority status of Jain religion.[web 10][note 31]
However, some individual states have over the past few decades differed on whether Jains, Buddhists and Sikhs are religious minorities or not, by either pronouncing judgments or passing legislation. One example is the judgment passed by the Supreme Court in 2006, in a case pertaining to the state of Uttar Pradesh, which declared Jainism to be indisputably distinct from Hinduism, but mentioned that, "The question as to whether the Jains are part of the Hindu religion is open to debate. However, the Supreme Court also noted various court cases that have held Jainism to be a distinct religion.
Another example is the Gujarat Freedom of Religion Bill, that is an amendment to a legislation that sought to define Jains and Buddhists as denominations within Hinduism.[web 11] Ultimately on 31 July 2007, finding it not in conformity with the concept of freedom of religion as embodied in Article 25 (1) of the Constitution, Governor Naval Kishore Sharma returned the Gujarat Freedom of Religion (Amendment) Bill, 2006 citing the widespread protests by the Jains[web 12] as well as Supreme Court's extrajudicial observation that Jainism is a "special religion formed on the basis of quintessence of Hindu religion by the Supreme Court".[web 13]
Traditionally, an ashram (Sanskrit: ashrama or ashramam) is a spiritual hermitage or a monastery in Indian religions.Comparative religion
Comparative religion is the branch of the study of religions concerned with the systematic comparison of the doctrines and practices of the world's religions. In general the comparative study of religion yields a deeper understanding of the fundamental philosophical concerns of religion such as ethics, metaphysics, and the nature and forms of salvation. Studying such material is meant to give one a broadened and more sophisticated understanding of human beliefs and practices regarding the sacred, numinous, spiritual, and divine.In the field of comparative religion, a common geographical classification of the main world religions includes Middle Eastern religions (including Iranian religions), Indian religions, East Asian religions, African religions, American religions, Oceanic religions, and
classical Hellenistic religions.Divine presence
Divine presence, presence of God, Inner God, or simply presence is a concept in religion, spirituality, and theology that deals with the ability of a god or gods to be "present" with human beings.
According to some types of monotheism God is omnipresent.Major religious groups
The world's principal religions and spiritual traditions may be classified into a small number of major groups, although this is by no means a uniform practice. This theory began in the 18th century with the goal of recognizing the relative levels of civility in societies.Mun (religion)
Mun or Munism (also called Bongthingism) is the traditional polytheistic, animist, shamanistic, and syncretic religion of the Lepcha people. It predates the seventh century Lepcha conversion to Lamaistic Buddhism, and since that time, the Lepcha have practiced it together with Buddhism. Since the arrival of Christian missionaries in the nineteenth century, Mun traditions have been followed alongside that religion as well. The traditional religion permits incorporation of Buddha and Jesus Christ as deities, depending on household beliefs.The exonym "Mun" derives from the traditional belief in spirits called mun or mung. Together with bongthing (also bungthing or bóngthíng), mun comprise a central element in the religion. These terms are also used to describe the shaman priesthood that officiates the respective spirits.The Mun religion and its priesthood are in decline. Conversion to other religions is attributed to economic pressure, as traditional practices are immensely expensive to the ordinary practitioner. It has, however, regained interest among Lepcha as ecological encroachment becomes a growing concern. The environment is so deeply intertwined with Mun beliefs that religious leaders have offered direct opposition to development in areas including the Rathong Chu and Teesta Rivers.Nandavarta
The Nandavarta or Nandyavarta is one of the eight auspicious symbols of Jainism for the Svetambara sect. It is an ashtamangala which is used for worship, and could be made with rice grains. It is also the symbol of 18th Tirthankar Aranatha. The symbol has 4 arms with compulsorily 9 corners/ turns each.Patrick Olivelle
Patrick Olivelle is an Indologist. A philologist and scholar of Sanskrit Literature whose work has focused on asceticism, renunciation and the dharma, Olivelle has been Professor of Sanskrit and Indian Religions in the Department of Asian Studies at the University of Texas, Austin since 1991.Olivelle was born in Sri Lanka. He received a B.A. (Honours) in 1972 from the University of Oxford, where he studied Sanskrit, Pali and Indian Religions with Thomas Burrow and R.C. Zaehner. He received his Ph.D. from the University of Pennsylvania in 1974 for a thesis containing the critical edition and translation of Yadava Prakasa's Yatidharmaprakasa. Between 1974 and 1991 Olivelle taught in the Department of Religious Studies at the Indiana University, Bloomington.Prakṛti
Prakriti or Prakruti (from Sanskrit language प्रकृति, prakṛiti), means "nature". It is a key concept in Hinduism, formulated by its Samkhya school, and refers to the primal matter with three different innate qualities (Guṇas) whose equilibrium is the basis of all observed empirical reality. Prakriti, in this school, contrasts with Purusha which is pure awareness and metaphysical consciousness. The term is also found in the texts of other Indian religions such as Jainism, and Buddhism.In Indian languages derived from Indo-European Sanskrit roots, Prakriti refers to the feminine aspect of all life forms, and more specifically a woman is seen as a symbol of Prakriti.Religion in Tanzania
Christianity is the largest religion in Tanzania according to the most recent estimates. There are also substantial Muslim and Animist minorities.
Current statistics on the relative sizes of various religions in Tanzania are limited because religious questions have been eliminated from government census reports since 1967. A 2014 survey from the Pew Research Center found that 61% of the population are Christian, 35% are Muslim, 2% practice traditional religions and 1% are unaffiliated.Sanamahism
Sanamahism or Sanamahi Laining refers to the traditional Meitei beliefs and religion found in the state of India called Manipur near Myanmar. The term is derived from Sanamahi, one of the important Meitei deities. Sanamahi derives his power from the combination of all the stars in the Milky Way. Lord Sun worships him for more power and he delivers all with ease. According to Bertil Lintner, Sanamahism is an "animistic, ancestor worshipping, shaman-led tradition".Sanamahism is practiced by the Meitei, Zeliangrong and other communities who inhabit Manipur, Assam, Tripura, Myanmar and Bangladesh, with small migrant populations in the United Kingdom, United States, and Canada.Sant (religion)
In Hinduism, Jainism and Buddhism, a sant is a human being revered for his or her knowledge of "self, truth, reality" and as a "truth-exemplar". In Sikhism it is used to describe a being who has attained spiritual enlightenment and divine knowledge and power through the union with the monotheistic God.Sarnaism
Sarnaism or Sarna (local languages: Sarna Dhorom or Sarna Dharam, meaning "religion of the woods"), also known as Sariism (Sari Dharam, literally "sal tree religion") or Adiism (Adi Dharam, literally "original religion"), is the indigenous religion of the Adivasi populations of the states of Jharkhand, Odisha, West Bengal, Bihar, Madhya Pradesh, Maharashtra and Chhattisgarh, centred around the worship of nature represented by trees. Followers of these religions primarily belong to the Munda, Ho, Bhumij, Santal,Baiga and Khuruk ethnic groups.Saṃsāra
Saṃsāra () is a Sanskrit word that means "wandering" or "world", with the connotation of cyclic, circuitous change. It also refers to the concept of rebirth and "cyclicality of all life, matter, existence", a fundamental assumption of most Indian religions. In short, it is the cycle of death and rebirth. Saṃsāra is sometimes referred to with terms or phrases such as transmigration, karmic cycle, reincarnation, and "cycle of aimless drifting, wandering or mundane existence".The concept of Saṃsāra has roots in the post-Vedic literature; the theory is not discussed in the Vedas themselves. It appears in developed form, but without mechanistic details, in the early Upanishads. The full exposition of the Saṃsāra doctrine is found in Sramanic religions such as Buddhism and Jainism, as well as various schools of Hindu philosophy after about the mid-1st millennium BCE. The Saṃsāra doctrine is tied to the Karma theory of Indian religions, and the liberation from Saṃsāra has been at the core of the spiritual quest of Indian traditions, as well as their internal disagreements. The liberation from Saṃsāra is called Moksha, Nirvana, Mukti or Kaivalya.Self-realization
Self-realization is an expression used in Western psychology, philosophy, and spirituality; and in Indian religions. In the Western understanding it is the "fulfillment by oneself of the possibilities of one's character or personality." In the Indian understanding, self-realization is liberating knowledge of the true Self, either as the permanent undying Atman, or as the absence (sunyata) of such a permanent Self.Sentience
Sentience is the capacity to feel, perceive or experience subjectively. Eighteenth-century philosophers used the concept to distinguish the ability to think (reason) from the ability to feel (sentience). In modern Western philosophy, sentience is the ability to experience sensations (known in philosophy of mind as "qualia"). In Eastern philosophy, sentience is a metaphysical quality of all things that require respect and care. The concept is central to the philosophy of animal rights because sentience is necessary for the ability to suffer, and thus is held to confer certain rights.Shrivatsa
The Shrivatsa (Sanskrit श्रीवत्स śrīvatsa) is an ancient symbol considered auspicious in Indian religious traditions.Tapas (Indian religions)
Tapas is a Sanskrit word that means "to heat". It also connotes certain spiritual practices in Indian religions. In Jainism, it refers to asceticism (austerities, body mortification); in Buddhism to spiritual practices including meditation and self-discipline; and in the different traditions within Hinduism it refers to a spectrum of practices ranging from asceticism, inner cleansing to self-discipline. The Tapas practice often involves solitude, and is a part of monastic practices that are believed to be a means to moksha (liberation, salvation).In the Vedas literature of Hinduism, fusion words based on tapas are widely used to expound several spiritual concepts that develop through heat or inner energy, such as meditation, any process to reach special observations and insights, the spiritual ecstasy of a yogin or Tāpasa (a vṛddhi derivative meaning "a practitioner of austerities, an ascetic"), even warmth of sexual intimacy. In certain contexts, the term means penance, pious activity, as well as severe meditation.Tribal religions in India
About 104 million people in India are members of Scheduled Tribes, which accounts for 8.6 % of India's population (according to the 2011 census). In the census of India from 1871 to 1941, tribals have been counted in different religions from other religions,1891(forest tribe), 1901(animist),1911(tribal animist), 1921(hill and forest tribe), 1931(primitive tribe), 1941(tribes), However, since the census of 1951, the tribal population has been stopped separately.Now many Indians belonging to these populations adhere to traditional Indian tribal religions, often syncretised with one or more of the major religious traditions of Hinduism, Buddhism, Islam and/or Christianity and often under ongoing pressure of cultural assimilation.The tribal people observe their festivals, which have no direct conflict with any religion, and they conduct marriage among them according to their tribal custom.
They have their own way of life to maintain all privileges in matters connected with marriage and succession, according to their customary tribal faith.
In keeping with the nature of Indian religion generally, these particular religions often involve traditions of ancestor worship or worship of spirits of natural features. Tribal beliefs persist as folk religion even among those converted to a major religion.
The largest and best-known tribal religion of India is that of the Santhal of Orissa.
In 1991, there were some 24,000 Indians belonging to the Santhal community who identified explicitly as adherents of the Santhal traditional religion in the Indian census, as opposed to 300,000 who identified as Christians. Among the Munda people and Oraons of Bihar, about 25 % of the population are Christian. Among the Kharia people of Bihar (population about 130,000), about 60 % are Christians, but all are heavily influenced by Folk Hinduism. Tribal groups in the Himalayas were similarly affected by both Hinduism and Buddhism in the late 20th century. The small hunting-and-gathering groups in the union territory of the Andaman and Nicobar Islands have also been under severe pressure of cultural assimilation.<|endoftext|>
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## Pollard’s Rho Algorithm For Discrete Logarithms
### May 27, 2016
We studied discrete logarithms in two previous exercises. Today we look at a third algorithm for computing discrete algorithms, invented by John Pollard in the mid 1970s. Our presentation follows that in the book Prime Numbers: A Computational Perspective by Richard Crandall and Carl Pomerance, which differs somewhat from other sources.
Our goal is to compute l (some browsers mess that up; it’s a lower-case ell, for “logarithm”) in the expression glt (mod p); here p is a prime greater than 3, g is an integer generator on the range 1 ≤ g < p, and t is an integer target on the range 1 ≤ g < p. Pollard takes a sequence of integer pairs (ai, bi) modulo (p − 1) and a sequence of integers xi modulo p such that xi = tai gbi (mod p), beginning with a0 = b0 = 0 and x0 = 1. Then the rule for deriving the terms of the various sequences is:
• If 0 < xi < p/3, then ai+1 = (ai + 1) mod (p − 1), bi+1 = bi, and xi+1 = t xi (mod p).
• If p/3 < xi < 2p/3, then ai+1 = 2 ai mod (p − 1), bi+1 = 2 bi mod (p − 1), and xi+1 = xi2 mod p.
• If 2p/3 < xi < p, then ai+1 = ai, bi+1 = (bi + 1) mod (p − 1), and xi+1 = g xi mod p.
Splitting the computation into three pieces “randomizes” the calculation, since the interval in which xi is found has nothing to do with the logarithm. The sequences are computed until some xj = xk, at which point we have taj gbj = tak gbk. Then, if ajaj is coprime to p − 1, we compute the discrete logarithm l as (ajak) lbkbj (mod (p − 1)). However, if the greatest common divisor of ajaj with p − 1 is d > 1, then we compute (ajak) l0bkbj (mod (p − 1) / d), and l = l0 + m (p − 1) / d for some m = 0, 1, …, d − 1, which must all be checked until the discrete logarithm is found.
Thus, Pollard’s rho algorithm consists of iterating the sequences until a match is found, for which we use Floyd’s cycle-finding algorithm, just as in Pollard’s rho algorithm for factoring integers. Here are outlines of the two algorithms, shown side-by-side to highlight the similarities:
```# find d such that d | n # find l such that g**l = t (mod p)
function factor(n) function dlog(g, t, p)
func f(x) := (x*x+c) % n func f(x,a,b) := ... as above ...
t, h, d := 1, 1, 1 j := (1,0,0); k := f(1,0,0)
while d == 1 while j.x <> k.x
t = f(t) j(x,a,b) := f(j.x, j.a, j.b)
h = f(f(h)) k(x,a,b) := f(f(k.x, k.a, k.b))
d = gcd(t-h, n) d := gcd(j.a-k.a, p-1)
return d return l ... as above ...```
Please pardon some abuse of notation; I hope the intent is clear. In the factoring algorithm, it is possible that d is the trivial factor n, in which case you must try again with a different constant in the f function; the logarithm function has no such possibility. Most of the time consumed in the computation is the modular multiplications in the calculations of the x sequence; the algorithm itself is O(sqrt p), the same as the baby-steps, giant-steps algorithm of a previous exercise, but the space requirement is only a small constant, rather than the O(sqrt p) space required of the previous algorithm. In practice, the random split is made into more than 3 pieces, which complicates the code but speeds the computation, as much as a 25% improvement on average.
Your task is to write a program that computes discrete logarithms using Pollard’s rho algorithm. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.<|endoftext|>
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A typical crawling pattern would be considered crawling on hands and knees symmetrically. Most children crawl between the ages of around 6-10 months. Most children progress from belly crawling/combat crawling to hand/knee crawling. Crawling is a very important motor milestone that should not be skipped. Some children deviate from the typical crawling pattern in a variety of ways, and it doesn’t always indicate that anything wrong, but can be a red flag for developmental delay or even a diagnosis. One major red flag to look for in crawling patterns would be asymmetry, for example a child who uses one side of their body unequally to the other ( for example, right arm and right leg pulls body forward, while left arm and leg lag). Some children “bear crawl” on hands and feet, not letting the knees touch the floor, and while this looks unusual, it is not always abnormal. Some children will “crab crawl” on one knee, while keeping the other knee bent and the foot flat on the floor to propel themselves.
Some children scoot on their bottoms, propelling themselves with their hands instead of crawling. Some children will roll to get where they want instead of crawling. Children with low or high muscle tone can find crawling difficult to master, and also babies who did not spend time on their tummy’s or have weak core strength and or poor head/neck control or tight hip flexors can find crawling difficult. Also, children with sensory concerns may dislike crawling, not being able to tolerate the feel of carpeting on their knees or hands as they weight bear. The best suggestion is to have a a child evaluated by an early intervention physical therapist before age one if there are any concerns about a child’s crawling pattern or lack of crawling.<|endoftext|>
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Today is Bill of Rights Day. On this day, December 15, in 1791, the United States adopted the first ten amendments to the Constitution, securing critical checks on the new central government.
“Liberty must at all hazards be supported. We have a right to it, derived from our Maker. But if we had not, our fathers have earned and bought it for us, at the expense of their ease, their estates, their pleasure, and their blood.” – John Adams, 1765
“The powers delegated by the proposed Constitution to the federal government are few and defined. Those which are to remain in the State governments are numerous and indefinite.” – James Madison, Federalist 45, 1788
“A wise and frugal government, which shall leave men free to regulate their own pursuits of industry and improvement, and shall not take from the mouth of labor the bread it has earned — this is the sum of good government.” – Thomas Jefferson, First Inaugural Address, 1801
Without an agreement that a Bill of Rights would be added, the Constitution would not have been ratified. These ten amendments guarantee the freedoms and rights that proceeded to allow the United States the great nation it became.
The Bill of Rights
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the
A well-regulated militia, being necessary to the security of a free state, the right of the people to keep and bear arms, shall not be infringed.
No soldier shall, in time of peace be quartered in any house, without the consent of the owner, nor in time of war, but in a manner to be prescribed by law.
The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no warrants shall issue, but upon probable cause, supported by oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized.
No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a grand jury, except in cases arising in the land or naval forces, or in the militia, when in actual service in time of war or public danger;nor shall any person be subject for the same offense to be twice put in jeopardy of life or limb; nor shall be compelled in any criminal case to be a witness against himself, nor be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation.
In all criminal prosecutions, the accused shall enjoy the right to a speedy and public trial, by an impartial jury of the state and district wherein the crime shall have been committed, which district shall have been previously ascertained by law, and to be informed of the nature and cause of the accusation; to be confronted with the witnesses against him; to have compulsory process for obtaining witnesses in his favor, and to have the assistance of counsel for his defense.
In suits at common law, where the value in controversy shall exceed twenty dollars, the right of trial by jury shall be preserved, and no fact tried by a jury, shall be otherwise reexamined in any court of the United States,
Excessive bail shall not be required, nor excessive fines imposed, nor cruel and unusual punishments inflicted.
The enumeration in the Constitution, of certain rights, shall not be construed to deny or disparage others retained by the people.
The powers not delegated to the United States by the Constitution, nor prohibited by it to the states, are reserved to the states respectively, or to the people.<|endoftext|>
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12 December
These three vertices form a right angled triangle.
There are 2600 different ways to pick three vertices of a regular 26-sided shape. Sometime the three vertices you pick form a right angled triangle.
Today's number is the number of different ways to pick three vertices of a regular 26-sided shape so that the three vertices make a right angled triangle.
Is it equilateral?
In the diagram below, $$ABDC$$ is a square. Angles $$ACE$$ and $$BDE$$ are both 75°.
Is triangle $$ABE$$ equilateral? Why/why not?
20 December
Earlier this year, I wrote a blog post about different ways to prove Pythagoras' theorem. Today's puzzle uses Pythagoras' theorem.
Start with a line of length 2. Draw a line of length 17 perpendicular to it. Connect the ends to make a right-angled triangle. The length of the hypotenuse of this triangle will be a non-integer.
Draw a line of length 17 perpendicular to the hypotenuse and make another right-angled triangle. Again the new hypotenuse will have a non-integer length. Repeat this until you get a hypotenuse of integer length. What is the length of this hypotenuse?
Cutting corners
The diagram below shows a triangle $$ABC$$. The line $$CE$$ is perpendicular to $$AB$$ and the line $$AD$$ is perpedicular to $$BC$$.
The side $$AC$$ is 6.5cm long and the lines $$CE$$ and $$AD$$ are 5.6cm and 6.0cm respectively.
How long are the other two sides of the triangle?
Two triangles
Source: Maths Jam
The three sides of this triangle have been split into three equal parts and three lines have been added.
What is the area of the smaller blue triangle as a fraction of the area of the original large triangle?
Equal side and angle
In the diagram shown, the lengths $$AD = CD$$ and the angles $$ABD=CBD$$.
Prove that the lengths $$AB=BC$$.
Arctan
Prove that $$\arctan(1)+\arctan(2)+\arctan(3)=\pi$$.
Square deal
This unit square is divided into four regions by a diagonal and a line that connects a vertex to the midpoint of an opposite side. What are the areas of the four regions?
© Matthew Scroggs 2018<|endoftext|>
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While multiplying three real numbers, Ashok took one of the numbers as $73$ instead of $37$. As a result, the product went up by $720$. Then the minimum possible value of the sum of squares of the other two numbers is _________
Let the three numbers be $x,y$ and $z.$
Let, ashok took $z = 73$ instead of $z = 37$
So, the final product will be
$xy(73) – xy(37) = 720$
$\Rightarrow xy(73 -37) = 720$
$\Rightarrow xy(36) = 720$
$\Rightarrow \boxed {xy = 20} \quad \longrightarrow (1)$
We know that, $\text{AM}$ and $\text{GM}$ are arithmetic and geometric mean respectively between two numbers $x$ and $y$, then
$\boxed {\text{AM} \geqslant \text{GM}}$
$\Rightarrow \frac{(x+y)}{2} \geqslant \sqrt{xy}$
$\Rightarrow x+y \geqslant 2 \sqrt{xy}$
$\Rightarrow (x+y)^{2} \geqslant 4xy$
$\Rightarrow x^{2} + y^{2} + 2xy \geqslant 4xy$
$\Rightarrow x^{2} + y^{2} \geqslant 2xy$
$\Rightarrow x^{2} + y^{2} \geqslant 2(20)$
$\Rightarrow \boxed {x^{2} + y^{2} \geqslant 40}$
$\textbf{Short Method:}$
The minimum possible sum of the squares of the two numbers when $x = y.$
From equation $(1), x = y = \sqrt {20}$
so, $x^{2} + y^{2} = (\sqrt {20})^{2} + ( \sqrt {20})^{2}$
$\Rightarrow x^{2} + y^{2} = 20 + 20 = 40$
$\therefore$ The minimum possible value of the sum of squares of the other two numbers is $40.$
Correct Answer $:40$
10.3k points
1
285 views<|endoftext|>
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# Lesson 3: Circle Square Patterns
## Overview
Students will create rules for ordering patterns of circles and squares. Students generate all possible messages with three place values, then create rules that explain how they ordered each message. Emphasis is placed on creating clear rules so that, if another group were to follow the rules, they would generate the same list in the same order. Using these rules, students then try to list all possible messages with four place values. As the lesson concludes, students share their rules with classmates.
## Purpose
Eventually, students will need to understand the binary number system which uses 1's and 0's rather than circles and squares. This lesson acts as a bridge to the next lesson where binary is formally introduced and practiced. In wrestling with the challenge of describing the rules of ordering patterns of circles and squares clearly, students will be primed to see how the binary number system solves many of these problems. Students may discover a system that is equivalent to the binary number system, which is a feat worth celebrating, but it is not expected that every student uncovers the rules for binary in this lesson.
## Objectives
### Students will be able to:
• Follow a set of rules for ordering sets of patterns
• Explain the challenges of creating a clear set of rules for ordering patterns
## Preparation
Heads Up! Please make a copy of any documents you plan to share with students.
# Teaching Guide
## Lesson Modifications
Attention, teachers! If you are teaching virtually or in a socially-distanced classroom, please read the full lesson plan below, then click here to access the modifications.
## Warm Up (5 mins)
Discussion Goal
Goal: Introduce the idea that the numeral '7' is just one commonly used symbol to represent the number seven. There are many ways to communicate this same number that may use different symbols or representations, all of which are valid. Sharing the variety of responses helps motivate the following activity that asks students to discover a system for using the circles and squares to represent numbers, including the number 7.
Prompt: How many ways can you represent 7?
Discuss: Students should brainstorm individually before sharing in small groups. It is important to allow ample brainstorming time - students may generate familiar responses at first, but may stretch their thinking and get more creative with additional time. Some examples may include:
• Linguistic examples - "seven", "siete" (spanish), "sept" (french), "sieben" (german), etc
• Picture examples - dots, tallys, emojis, etc
• Math & Geometry examples - 5 + 2, 8 - 1, a seven-sided shape, etc
After a short time, ask students to share some responses with the whole class. Use these responses to quickly generate a wide variety of representations.
Remarks
There are a variety of ways we could represent the number 7 - we might use the numeral '7' or the word 'seven', but this might be different in other countries or other languages. Today we'll see how we might represent the number 7 using only two different shapes.
## Activity (35 mins)
### Generate Patterns (5 minutes)
Remarks
In the previous class, we ended by deciding that one of the best way to use our devices was to limit them to two options, let's say: option 1 is circles and option 2 is squares. Now let's figure out how we can use these shapes to communicate lots of different pieces of information.
Prompt: With a partner, work out how many different pieces of information (made of up of circles and squares) you can represent with three place values. For example: `circle-circle-circle` and `circle-square-circle` can represent two different pieces of information.
Discuss: Give students time to work individually, then have them share with their neighbor and fill in any patterns they may have missed.
Do This: Confirm with the class that there are 8 possible patterns, but don't list all of them out. Ask students to share the 7th pattern in their list. Students will likely have different answers for this.
Teaching Tip
Manipulatives: Students are given manipulatives to help visualize any rules they are using to move from one element of the list to the next. You might see students use all their manipulatives at once to create the different patterns, then discuss how to arrange them into an ordered list. You might see students representing one pattern at a time, then discussing the rules for “replacing” shapes to generate each of the next patterns. Students might not use the manipulatives at all, using pen & paper or whiteboards instead.
Many Possible Answers: It is okay for different groups to come up with different orders for their lists of patterns - this will help with the share-out discussion as you highlight different strategies.
Facilitating With Groups: You should act as a facilitator during this part of the activity, guiding students in describing the rules & strategies that they used to create their list. These strategies may be implicit and unconscious to the student, but you can ask questions to help students realize their own thinking that went into generating their list. Aim to help students clarify their thinking to make it easier for other groups to follow.
Group Dynamics: Be mindful of groups that appear to be dominated by a single student. Asking each student individually about their strategy can help bring students back together and reinforce the collaborative aspect of this activity.
Remarks
We agreed that there are 8 possible patterns we can make with 3 place values. But, not everyone wrote these patterns in the same order, which means we don't all have the same 7th pattern! Our goal is to create a clear set of rules where, if the class were to follow these rules, everyone should generate the same list of patterns in the same order.
### Circle Square Activity (25 minutes)
Group: Place students in groups of 2, making one group of 3 if necessary
Distribute: Circle Square Patterns - Activity Guide - one for each group. Each group also gets the Shape Cutouts - Resource to use during the activity.
Challenge #1: Students again list all possible three place value patterns, but with an added focus on the order of their patterns.
Challenge #2: Students describe the rules or strategy they used to create their list. They are aiming for clear directions that can be followed by another group to reproduce their same list in the same order.
Challenge #3: Students extend their rules to generate all possible four place value patterns. This challenge has 2 parts: discovering all possible four place value combinations (there are 16) and extending the rules from the previous challenge so they work here as well. Most groups will need to change or add to their rules in order to accomplish Challenge #3.
Teaching Tip
Goal: There are many ways to structure this discussion, especially if you have your own established share-out routines. Here are a few that could work for this particular discussion:
• Have each group trade with another group, and each group tries to re-create the original group's list. This strategy is useful if you have more class time than expected during the wrap-up.
• A group reads their rules while you and the rest of the class try to recreate the list
• A group reveals their list and the class tries to predict what the rules are, then the group shares their rules with the class
• You can name different strategies you’ve seen from groups as you’ve been circulating, then ask groups to give a thumbs-up if their rules involved that particular pattern. This strategy is useful if you’re running short on time before the next part of the lesson.
Discuss: Select a few groups to share out their rules, highlighting groups with different strategies and rules they used for their lists. Emphasize the 7th item in each list, connecting it back to the warm-up activity as another way to represent the number 7.
## Wrap Up (5 mins)
Remarks
Congratulations! You just invented your own system for counting and we now have new ways to represent the number 7! This happens a lot as new technology is invented and fine-tuned - different technologies might count in different ways. Tomorrow we're going to learn about the counting system computers use to represent numbers!
Discussion Goal
Similar:
• It follows agreed upon rules
Different:
• Uses shapes rather than numbers
Prompt: How is counting in this circle/square system similar to how we count in our regular lives? How is it different?
### Assessment: Check For Understanding
Check For Understanding Question(s) and solutions can be found in each lesson on Code Studio. These questions can be used for an exit ticket.
Question: How would you explain a number system to someone who had never seen numbers before?
• 2
• 3
• (click tabs to see student view)
View on Code Studio
### Student Instructions
View on Code Studio
### Student Instructions
In 50 words or less, describe the concept of a number system.
Why are rules required for a number system to be useful?
## Standards Alignment
#### CSTA K-12 Computer Science Standards (2017)
DA - Data & Analysis
• 3A-DA-09 - Translate between different bit representations of real-world phenomena, such as characters, numbers, and images.<|endoftext|>
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Go outside on a dark, moonless night. Look up. Is it December or January? Check out Betelgeuse, glowing dully red at Orion’s shoulder, and Rigel, a laser blue at his knee. A month later, yellow Capella rides high in Auriga.
Is it July? Find Vega, a sapphire in Lyra, or Antares, the orange-red heart of Scorpius.
In fact, any time of the year you can find colors in the sky. Most stars look white, but the brightest ones show color. Red, orange, yellow, blue… almost all the colors of the rainbow. But hey, wait a sec. Where are the green stars? Shouldn’t we see them?
Nope. It’s a very common question, but in fact we don’t see any green stars at all. Here’s why.
Take a blowtorch (figuratively!) and heat up an iron bar. After a moment it will glow red, then orange, then bluish-white. Then it’ll melt. Better use a pot holder.
Why does it glow? Any matter above the temperature of absolute zero (about -273 Celsius) will emit light. The amount of light it gives off, and more importantly the wavelength of that light, depends on the temperature. The warmer the object, the shorter the wavelength.
Cold objects emit radio waves. Extremely hot objects emit ultraviolet light, or X-rays. At a very narrow of temperatures, hot objects will emit visible light (wavelengths from roughly 300 nanometers to about 700 nm).
Mind you — and this is critical in a minute — the objects don’t emit a single wavelength of light. Instead, they emit photons in a range of wavelengths. If you were to use some sort of detector that is sensitive to the wavelengths of light emitted by an object, and then plotted the number of them versus wavelength, you get a lopsided plot called a blackbody curve (the reason behind that name isn’t important here, but you can look it up if you care — just set your SafeSearch Filtering to "on". Trust me here). It’s a bit like a bell curve, but it cuts off sharply at shorter wavelengths, and tails off at longer ones.
Here’s an example of several curves, corresponding to various temperatures of objects (taken from online lecture notes at UW:
The x-axis is wavelength (color, if you like) color, and the spectrum of visible colors is superposed for reference. You can see the characteristic shape of the blackbody curve. As the object gets hotter, the peak shifts to the left, to shorter wavelengths.
An object that is at 4500 Kelvins (about 4200 Celsius or 7600 F) peaks in the orange part of the spectrum. Warm it up to 6000 Kelvin (about the temperature of the Sun, 5700 C or 10,000 F) and it peaks in the blue-green. Heat it up more, and the peaks moves into the blue, or even toward shorter wavelengths. In fact, the hottest stars put out most of their light in the ultraviolet, at shorter wavelengths than we can see with our eyes.
Now wait a sec (again)… if the Sun peaks in the blue-green, why doesn’t it look blue-green?
Ah, this is the key question! It’s because it might peak in the blue-green, but it still emits light at other colors.
Look at the graph for an object as hot as the Sun. That curve peaks at blue-green, so it emits most of its photons there. But it still emits some that are bluer, and some that are redder. When we look at the Sun, we see all these colors blended together. Our eyes mix them up to produce one color: white. Yes, white. Some people say the Sun is yellow, but if it were really yellow to our eyes, then clouds would look yellow, and snow would too (all of it, not just some of it in your back yard where your dog hangs out).
OK, so the Sun doesn’t look green. But can we fiddle with the temperature to get a green star? Maybe one that’s slightly warmer or cooler than the Sun?
It turns out that no, you can’t. A warmer star will put out more blue, and a cooler one more red, but no matter what, our eyes just won’t see that as green.
The fault lies not in the stars (well, not entirely), but within ourselves.
Our eyes have light-sensitive cells in them called rods and cones. Rods are basically the brightness detectors, and are blind to color. Cones see color, and there are three kinds: ones sensitive to red, others to blue, and the third to green. When light hits them, each gets triggered by a different amount; red light (say, from a strawberry) really gets the red cones juiced, but the blue and green cones are rather blasé about it.
Most objects don’t emit (or reflect) one color, so the cones are triggered by varying amounts. An orange, for example, gets the red cones going about twice as much as the green ones, but leaves the blue ones alone. When the brain receives the signal from the three cones, it says "This must be an object that is orange." If the green cones are seeing just as much light as the red, with the blue ones not seeing anything, we interpret that as yellow. And so on.
So the only way to see a star as being green is for it to be only emitting green light. But as you can see from the graph above, that’s pretty much impossible. Any star emitting mostly green will be putting out lots of red and blue as well, making the star look white. Changing the star’s temperature will make it look orange, or yellow, or red, or blue, but you just can’t get green. Our eyes simply won’t see it that way.
That’s why there are no green stars. The colors emitted by stars together with how our eyes see those colors pretty much guarantees it.
But that doesn’t bug me. If you’ve ever put your eye to a telescope and seen gleaming Vega or ruddy Antares or the deeply orange Arcturus, you won’t mind much either. Stars don’t come in all colors, but they come in enough colors, and they’re fantastically beautiful because of it.
Note: this is not the end of the story. There are green objects in space, and some stars do appear green… but that’s for another post, coming soon. Promise.<|endoftext|>
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Subsections
# Polar Coordinates
## Background
The use of polar coordinates allows for the analysis of families of curves difficult to handle through rectangular coordinates. If a curve is a rectangular coordinate graph of a function, it cannot have any loops since, for a given value there can be at most one corresponding value. However, using polar coordinates, curves with loops can appear as graphs of functions
## Plotting Polar Curves
When you graph curves in polar coordinates, you are really working with parametric curves. The basic idea is that you want to plot a set of points by giving their coordinates in pairs. When you use polar coordinates, you are defining the points in terms of polar coordinates . When you plot polar curves, you are usually assuming that is a function of the angle and is the parameter that describes the curve. In Maple you have to put square brackets around the curve and add the specification coords=polar. Maple assumes that the first coordinate in the parametric plot is the radius and the second coordinates is the angle .
### Cardioids, Limaçons, and Roses
These are three types of well-known graphs in polar coordinates. The table below will allow you to identify the graphs in the exercises.
Name Equation cardioid or limaçon or rose or
Below is an example of a cardiod.
>plot(1-cos(theta),theta=0..2*Pi,coords=polar);
## Area in Polar Coordinates
The relationship between area and integrals in polar coordinates is a little strange; the area inside a circle given (in polar coordinates) by is NOT just . Here is the rule: Area inside is given by . This comes from the fact that the area in a thin wedge with radius and angle is . Note that this gives you the right answer for a circle: . So to find the area of the cardiod use the following command.
>Area1:=1/2*int((1-cos(theta))^2, theta=0..2*Pi);
>evalf(Area1);
## Exercises
1. For each of the following polar equations, plot the graph in polar coordinates using the plot command and identify the graph as a cardioid, limaçon, or rose.
2. Find all points of intersection for each pair of curves in polar coordinates.
1. and for .
2. and for .
3. Find the angles that create only one petal of the five petal rose given by the equation . Plot only one petal and find the area of that petal.<|endoftext|>
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A server in computing can serve a number of different purposes. A single server is able to share data among many different client. Alternatively, many servers can be used discretely by one client for performing various computations or sharing with thousands of different clients. All of this is made possible by the computational architecture powering servers.
The Architecture of Servers
In the most technical sense, a server is a program but, more typically, a device which provides enhanced functionality for one or more other programs and devices. In computing, these other programs and/or devices are called clients. The underlying architecture empowering servers is characterized as the client-server model.
Different Kinds of Servers
There are many different kinds of servers, which serve subtly different functions. These different kind of servers include: database servers, web servers, application servers, file servers, mail servers, and print servers. The commonality across these different is a dedication to the client-server model.
The client-server model is a distributed application structure designed to designate workloads between providers of a resource (i.e., the servers) and the requester of that service (i.e., the client). The service, strictly speaking, may be a resource or some type of data.
Servers and Networks
A server is a computer designed with the purpose of processing requests and delivering data to one or more other computers over a local network or the internet. The word “server” is today used most in conjunction with the internet, where most people think of the term “server” in the context of a web server.
A web server allows web pages to be accessed over the internet thanks to the server in the client-server model hosting that material. In a more limited sense, local servers such as file servers merely store data rather than curate that data across a network for the entire internet to enjoy.
Conclusively, servers work within the client-server model and come in many different types.<|endoftext|>
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Trains : PS Archive
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# Trains
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It takes the high-speed train x hours to travel the z miles from Town A to Town B at a constant rate, while it takes the regular train y hours to travel the same distance at a constant rate. If the high-speed train leaves Town A for Town B at the same time that the regular train leaves Town B for Town A, how many more miles will the high-speed train have traveled than the regular train when the two trains pass each other?
A. z(y – x)/x + y
B. z(x – y)/x + y
C. z(x + y)/y – x
D. xy(x – y)/x + y
E. xy(y – x)/x + y
I followed this approach: Since trains are approaching each other, relative speed =x+y
Let t be the time taken then t= z/(x+y)
Now distance travelled by high speed train = t*x=zx/(x+y)
distance travelled by average speed train = t*y=zy/(x+y)
Hence difference between both = z(x-y)/(x+y)
But the OA is z(y-x)/(x+y)
What is wrong in my approach?
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03 Apr 2009, 03:25
Your t isn't correct. x and y are not the speeds of the trains. Hint: The speed of the high-speed train is z/x.
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### Show Tags
03 Apr 2009, 05:20
Oh my god! In a hurry, i misread vthe time as speed. Bad day!
Raison wrote:
Your t isn't correct. x and y are not the speeds of the trains. Hint: The speed of the high-speed train is z/x.
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03 Apr 2009, 15:20
I figured that's what happened. It happens to all of us at times.
Re: Trains [#permalink] 03 Apr 2009, 15:20
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Balbharti Maharashtra State Board Class 9 Maths Solutions covers the Practice Set 9.1 Geometry 9th Class Maths Part 2 Answers Solutions Chapter 9 Surface Area and Volume.
## Practice Set 9.1 Geometry 9th Std Maths Part 2 Answers Chapter 9 Surface Area and Volume
Question 1.
Length, breadth and height of a cuboid shape box of medicine is 20 cm, 12 cm and 10 cm respectively. Find the surface area of vertical faces and total surface area of this box.
Given: For cuboid shape box of medicine,
length (l) = 20 cm, breadth (b) = 12 cm and height (h) = 10 cm.
To find: Surface area of vertical faces and total surface area of the box
Solution:
i. Surface area of vertical faces of the box
= 2(l + b) x h
= 2(20+ 12) x 10
= 2 x 32 x 10
= 640 sq.cm.
ii. Total surface area of the box
= 2 (lb + bh + lh)
= 2(20 x 12+ 12 x 10 + 20 x 10)
= 2(240 + 120 + 200)
= 2 x 560
= 1120 sq.cm.
∴ The surface area of vertical faces and total surface area of the box are 640 sq.cm, and 1120 sq.cm, respectively.
Question 2.
Total surface area of a box of cuboid shape is 500 sq.unit. Its breadth and height is 6 unit and 5 unit respectively. What is the length of that box?
Given: For cuboid shape box,
breadth (b) = 6 unit, height (h) = 5 unit Total surface area = 500 sq. unit.
To find: Length of the box (l)
Solution:
Total surface area of the box = 2 (lb + bh + lh)
∴ 500 = 2 (6l + 6 x 5 + 5l)
∴ $$\frac { 500 }{ 2 }$$ = (11l + 30)
∴ 250= 11l + 30
∴ 250 – 30= 11l
∴ 220 = 11l
∴ 220 = l
∴ $$\frac { 220 }{ 11 }$$ = l
∴ l = 20 units
∴ The length of the box is 20 units.
Question 3.
Side of a cube is 4.5 cm. Find the surface area of all vertical faces and total surface area of the cube.
Given: Side of cube (l) = 4.5 cm
To find: Surface area of all vertical faces and the total surface area of the cube
Solution:
i. Area of vertical faces of cube = 4l2
= 4 (4.5)2 = 4 x 20.25 = 81 sq.cm.
ii. Total surface area of the cube = 6l2
= 6 (4.5)2
= 6 x 20.25
= 121.5 sq.cm.
∴ The surface area of all vertical faces and the total surface area of the cube are 81 sq.cm, and 121.5 sq.cm, respectively.
Question 4.
Total surface area of a cube is 5400 sq. cm. Find the surface area of all vertical faces of the cube.
Given: Total surface area of cube = 5400 sq.cm.
To find: Surface area of all vertical faces of the cube
Solution:
i. Total surface area of cube = 6l2
∴ 5400 = 6l2
∴ $$\frac { 5400 }{ 6 }$$ = l2
∴ l2 = 900
ii. Area of vertical faces of cube = 4l2
= 4 x 900 = 3600 sq.cm.
∴ The surface area of all vertical faces of the cube is 3600 sq.cm.
Question 5.
Volume of a cuboid is 34.50 cubic metre. Breadth and height of the cuboid is 1.5 m and 1.15 m respectively. Find its length.
Given: Breadth (b) = 1.5 m, height (h) = 1.15 m
Volume of cuboid = 34.50 cubic metre
To find: Length of the cuboid (l)
Solution:
Volume of cuboid = l x b x h
∴ 34.50 = l x b x h
∴ 34.50 = l x 1.5 x 1.15
= 20
∴ The length of the cuboid is 20 m.
Question 6.
What will be the volume of a cube having length of edge 7.5 cm ?
Given: Length of edge of cube (l) = 7.5 cm
To find: Volume of a cube
Solution:
Volume of a cube = l2
= (7.5)3
= 421.875 ≈ 421.88 cubic cm
∴The volume of the cube is 421.88 cubic cm.
Question 7.
Radius of base of a cylinder is 20 cm and its height is 13 cm, find its curved surface area and total surface area, (π = 3.14)
Given: Radius (r) = 20 cm, height (h) = 13 cm
To find: Curved surface area and
the total surface area of the cylinder
Solution:
i. Curved surface area of cylinder = 2πrh
= 2 x 3.14 x 20 x 13
= 1632.8 sq.cm
ii. Total surface area of cylinder = 2πr(r + h)
= 2 x 3.14 x 20(20 + 13)
= 2 x 3.14 x 20 x 33 = 4144.8 sq.cm
∴ The curved surface area and the total surface area of the cylinder are 1632.8 sq.cm and 4144.8 sq.cm respectively.
Question 8.
Curved surface area of a cylinder is 1980 cm2 and radius of its base is 15 cm. Find the height of the cylinder. (π = $$\frac { 22 }{ 7 }$$)
Given: Curved surface area of cylinder = 1980 sq.cm., radius (r) = 15 cm
To find: Height of the cylinder (h)
Solution:
Curved surface area of cylinder = 2πrh
∴ 1980 = 2 x $$\frac { 22 }{ 7 }$$ x 15 x h
∴ $$h=\frac{1980 \times 7}{2 \times 22 \times 15}$$
∴ h = 21 cm
∴ The height of the cylinder is 21 cm.<|endoftext|>
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Science and Technology Facilities Council
|Printable version||E-mail this to a friend|
UK scientists use ALMA to spot supernova dust factory 160,000 light years away
UK scientists have used the ALMA telescope to help capture the remains of a recent supernova - or exploding star – that is brimming with freshly formed dust 160,000 light years from Earth. Supernovae are thought to produce a large amount of the dust within galaxies, especially in the early Universe, but direct evidence of their ability to create dust has been limited - until now. The results appear in Astrophysical Journal Letters.
An international team of scientists including two from each of University College London and Keele University have used the European Southern Observatory (ESO)’s Atacama Large Millimeter/submillimeter Array (ALMA) telescope to observe the glowing remains of Supernova 1987A 2. The relatively young supernova is in the Large Magellanic Cloud, a dwarf galaxy orbiting the Milky Way about 160,000 light‐years from Earth. SN 1987A is the closest observed supernova explosion since Johannes Kepler's observation of a supernova inside the Milky Way in 1604.
"Really early galaxies are incredibly dusty and this dust plays a major role in the evolution of galaxies," said Mikako Matsuura of University College London (UCL). "Today we know dust can be created in several ways, but in the early Universe most of it must have come from supernovae. We finally have direct evidence to support that theory."
Astronomers predicted that as the gas cooled after the explosion, large amounts of dust would form as atoms of oxygen, carbon, and silicon bonded together in the cold central regions of the remnant. However, earlier observations of SN 1987A with infrared telescopes, made during the first 500 days after the explosion, detected only a small amount of hot dust.
With ALMA's unprecedented resolution and sensitivity, the research team was able to image the far more abundant cold dust, which glows brightly in millimetre and submillimetre light. The astronomers estimate that the remnant now contains about 25 percent the mass of the Sun in newly formed dust. They also found that significant amounts of carbon monoxide and silicon monoxide have formed.
Dr Jacco van Loon from the Lennard-Jones Laboratories, Keele University, said:
"Since we first detected cold dust in the direction of SN1987A with telescopes in space and on the Atacama Altiplano we have been using the most powerful observing facilities on Earth to zoom in on the exact location of this dust and separate its radiation from that of other sources of radio emission. It took the many dishes of the new ALMA observatory, that joined together to mimic a very large telescope, to finally confirm what astronomers hoped to find."
Remy Indebetouw, an astronomer at the National Radio Astronomy Observatory (NRAO) and the University of Virginia, both in Charlottesville, USA, added: “The new ALMA results, which are the first of their kind, reveal a supernova remnant chock full of material that simply did not exist a few decades ago."
Jacco van Loon said the discovery is very exciting for everyone – not just astronomers: "While supernovae signal the explosive destruction of stars, for the rest of the Universe they are a source of new stuff and energy; our lives would be very different without the chemical elements that were synthesized in supernovae throughout the history of the Universe. Grains are incredibly difficult to make in the vast emptiness of space - much different from our daily experience! If supernovae indeed make lots of them then this would have had very important and positive consequences for the eventual formation of the Sun and Earth as well."
Masa Lakicevic a PhD student at Keele University and Professor Michael J Barlow from UCL are the other two UK authors on the paper in Astrophysical Journal Letters which is entitled, ‘Dust Production and Particle Acceleration in Supernova 1987A Revealed with ALMA’.
STFC subscribes to ESO giving UK scientists access to ESO’s telescopes.
More information including links to the paper and images can be found on ESO’s website:<|endoftext|>
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Martin Luther King, Jr.
Preaching a philosophy of nonviolence, Dr. Martin Luther King Jr. became a catalyst for social change in the 1950s and 1960s. He galvanized people of all races to participate in boycotts, marches, and demonstrations against racial injustice. His moral leadership stirred the conscience of the nation and helped bring about the passage of the Civil Rights Act of 1964. In that same year he was awarded the Nobel Peace Prize. King continued his work for justice and equality until he was assassinated in 1968.
--Holt McDougal, Literature Grade 9
Analyze the Impact of Word Choice
Determine Author Purpose
delineate and Evaluate Argument
See Mrs. Roderick--The documents used are copyright protected and cannot be published on this site.<|endoftext|>
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# Linear Equation In Two Variable
This covers the basics of linear equation in two variable.
This covers the basics of linear equation in two variable.
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## Weitere Verwandte Inhalte
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### Linear Equation In Two Variable
1. 1. LINEAR EQUATION IN TWO VARIABLE
2. 2. Let’s start with the journey with some basics here: (+,+)(-,+) (-,-) (+,-) Learning the Cartesian sign is important . This is the origin The Quadrants
3. 3. The word LINEAR originates from: Today we will study the equation for lines. In a Graph, these equations are used To get the position of a line
4. 4. Linear equation in two variable THE STANDARD FORM: ax+by+c=0 Here, a and b are the constants that cannot be 0 Example: 47x+7y=9c can be zero
5. 5. There are 2 methods to solve A Pair of Linear equation 1 Graphical method 2 Algebraic method ax+by+c=0
6. 6. Lets solve some examples quickly: Q. We need to plot the diagram for 5x+4y+20=0 and check whether (0,-5) lies in it. STEP 1: Assume a value for x and find the value of y x y We will take three observation to plot the point. Lets take the points -1,0,1 for x When x=-1 5(-1)+4y+20=0 .’.-5+4y=-20 .’.4y=-20+5 y=-15/4=-3.75 When x=0 5(0)+4y+20=0 .’.4y=-20 .’.y=-20/4 y=-5 When x=-1 5(1)+4y+20=0 .’.5+4y=-20 .’.4y=-20-5 y=-25/4=-6.25 -1 -3.75 0 -5 1 -6.25
7. 7. x -1 0 1 y -3.75 -5 -6.25 From this observation We come to know that (0,-5) Lies on the line Now lets plot our graph This is our graph based on Cartesian sign Remember to label The points and the line (0,-5) (-1,-3.75) (1,-6.25) (0,0) And we are done!
8. 8. The standard form: ax+bx+c=0 x+y=5 x+y-5=0 3x+7y-66=0 8y-4x=-12 -4x+8y+12=0 2x+ 𝟐 𝟑 𝐲 = 𝟕 2x+ 𝟐 𝟑 𝐲 − 𝟕 =0 Lets do some quick activity: Determine the coefficints and the constants from the given expressions a b c 1 1 -5 3 7 -66 8 -4 12 2 𝟐 𝟑 -7
9. 9. Quick facts: For the equations like: x=n, where n can be any integer. The line on the graph will always be Parallel to y-axis Examples: X=-5 X=3 X=6 X=-2 They’re parallel to y-axis Y-axis X-axis
10. 10. Quick facts: For the equations like: y=n, where n can be any integer. The line on the graph will always be Parallel to x-axis Examples: y=-5 y=3 y=6 y=-2 They’re parallel to x-axis Y-axis X-axis
11. 11. THANK YOU<|endoftext|>
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Statistics
# 2.6Skewness and the Mean, Median, and Mode
Statistics2.6 Skewness and the Mean, Median, and Mode
Consider the following data set:
4, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 10
This data set can be represented by the following histogram. Each interval has width 1, and each value is located in the middle of an interval.
Figure 2.18
The histogram displays a symmetrical distribution of data. A distribution is symmetrical if a vertical line can be drawn at some point in the histogram such that the shape to the left and the right of the vertical line are mirror images of each other. The mean, the median, and the mode are each seven for these data. In a perfectly symmetrical distribution, the mean and the median are the same. This example has one mode (unimodal), and the mode is the same as the mean and median. In a symmetrical distribution that has two modes (bimodal), the two modes would be different from the mean and median.
The histogram for the data: 4, 5, 6, 6, 6, 7, 7, 7, 7, 8 is not symmetrical. The right-hand side seems chopped off compared to the left-hand side. A distribution of this type is called skewed to the left because it is pulled out to the left. A skewed left distribution has more high values.
Figure 2.19
The mean is 6.3, the median is 6.5, and the mode is seven. Notice that the mean is less than the median, and they are both less than the mode. The mean and the median both reflect the skewing, but the mean reflects it more so. The mean is pulled toward the tail in a skewed distribution.
The histogram for the data: 6, 7, 7, 7, 7, 8, 8, 8, 9, 10 is also not symmetrical. It is skewed to the right. A skewed right distribution has more low values.
Figure 2.20
The mean is 7.7, the median is 7.5, and the mode is seven. Of the three statistics, the mean is the largest, while the mode is the smallest. Again, the mean reflects the skewing the most.
To summarize, generally if the distribution of data is skewed to the left, the mean is less than the median, which is often less than the mode. If the distribution of data is skewed to the right, the mode is often less than the median, which is less than the mean.
Skewness and symmetry become important when we discuss probability distributions in later chapters.
## Example 2.32
### Problem
Statistics are used to compare and sometimes identify authors. The following lists show a simple random sample that compares the letter counts for three authors.
Terry: 7, 9, 3, 3, 3, 4, 1, 3, 2, 2
Davis: 3, 3, 3, 4, 1, 4, 3, 2, 3, 1
Maris: 2, 3, 4, 4, 4, 6, 6, 6, 8, 3
1. Make a dot plot for the three authors and compare the shapes.
2. Calculate the mean for each.
3. Calculate the median for each.
4. Describe any pattern you notice between the shape and the measures of center.
## Try It 2.32
Discuss the mean, median, and mode for each of the following problems. Is there a pattern between the shape and measure of the center?
a.
Figure 2.24
b.
The Ages at Which Former U.S. Presidents Died
4 6 9
5 3 6 7 7 7 8
6 0 0 3 3 4 4 5 6 7 7 7 8
7 0 1 1 2 3 4 7 8 8 9
8 0 1 3 5 8
9 0 0 3 3
Key: 8|0 means 80.
Table 2.30
c.
Figure 2.25
Order a print copy
As an Amazon Associate we earn from qualifying purchases.<|endoftext|>
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## 54) Bridging Ten Subtraction
Ten Plus Bonds: Strategy Adding On
### Mathematics
Develop the strategy of bridging ten to subtract, using adding on, for wholes up to 20.
For example, to solve 13 – 8
Prior knowledge of part-part-whole, bonds of 10 and place value partitioning using ten plus bonds is needed.
### Language
• bridge ten
• subtract
• equals
• right
Note, in this activity students are finding a missing part in a subtraction equation.
They are NOT completing “sums”. The “sum” is the whole, found by adding the parts.
The word ‘sum’ is often incorrectly used when the word ‘equation’ is meant. It is impossible to “complete a subtraction sum”.
## Differentiation
### A little easier
##### Counting order
Calculate each question in counting order, working clockwise around the spinner, from the 12 o’clock position. Do not flick the spinner. Students complete each type of equation:
• Subtract 9
• Subtract 8
• Subtract 7
Assist students to identify patterns of adding on 1, 2 or 3 depending on the part being subtracted.
##### Use Bond Blocks
If students have difficulty partitioning the part being taken away to bridge ten, use an additional block.
For example, to model 13 – 8
Some students find it conceptually easier to place the block representing the part being added on, that is the answer to the subtraction. Place this block on top of this block of the block that was used to bridge ten and the other block needed to make the part that is the answer.
• From a top view they can see the parts of 8 and 5.
• From a front view they can see the partition of 2 and 3.
##### Directional difficulties
Click to download a Comparison and Number line Language: Desk Visual to support students who have directional and sequencing difficulties.
### A little harder
##### Bridging ten subtraction strategy taking away
Play “Ten Plus Bonds: Bridging Ten Subtraction Strategy Adding On a little harder”. In this game students have to solve subtraction by bridging ten, to twenty, for wholes up to 30.
##### Reduce scaffolds
The blocks are a scaffold to support calculation. If students can calculate without some or all of the blocks encourage this. Below is a progression of reducing scaffolding, from most to least support, using bridging ten for 23-18.
##### Choosing equations
Instruct students to write one subtraction equation, with a whole less than 20, that can be solved efficiently using adding on (adding on to the known part). Students model this with Bond Blocks to explain their reasoning.
Repeat for subtraction solved using taking away.
Discuss which types of questions are more suited to each type of subtraction and why.
### Progression
In the next activity students develop another strategy for adding numbers to 20, partitioning using five plus bonds. Go to
##### Activity 55
Ten Plus Bonds: Partitioning Addition, Strategy Five Plus Bonds<|endoftext|>
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<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
# 5.2: Projectile Motion for an Object Launched at an Angle
Difficulty Level: At Grade Created by: CK-12
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In the case of the human cannonball shown, all the vector and gravitational calculations must be worked out perfectly before the first practice session. With this activity, you cannot afford trial and error – the first miss might be the last trial.
### Projectile Motion for an Object Launched at an Angle
When an object is projected from rest at an upward angle, its initial velocity can be resolved into two components. These two components operate independently of each other. The upward velocity undergoes constant downward acceleration which will result in it rising to a highest point and then falling backward to the ground. The horizontal motion is constant velocity motion and undergoes no changes due to gravity.The analysis of the motion involves dealing with the two motions independently.
Example Problem: A cannon ball is fired with an initial velocity of 100. m/s at an angle of 45° above the horizontal. What maximum height will it reach and how far will it fly horizontally?
Solution: The first step in the analysis of this motion is to resolve the initial velocity into its vertical and horizontal components.
\begin{align*}v_{\text{i}-\text{up}} = (100. \ \text{m/s})(\sin 45^\circ) = (100. \ \text{m/s})(0.707) = 70.7 \ \text{m/s}\end{align*}
\begin{align*}v_{\text{i}-\text{horizontal}} = (100. \ \text{m/s})(\cos 45^\circ) = (100. \ \text{m/s})(0.707) = 70.7 \ \text{m/s}\end{align*}
We will deal with the vertical motion first. The vertical motion is symmetrical. As the object rises to its highest point and then falls back down, it will travel the same distance in each direction, and take the same amount of time. This is often hard to accept, but the amount of time the object takes to come to a stop at its highest point is the same amount of time it takes to return to where it was launched from. Similarly, the initial velocity upward will be the same magnitude (opposite in direction) as the final velocity when it returns to its original height. There are several ways we could approach the upward motion. We could calculate the time it would take gravity to bring the initial velocity to rest. Or, we could calculate the time it would take gravity to change the initial velocity from +70.7 m/s to -70.0 m/s. Yet another way would be to calculate the time it takes for the object to return to its original height.
\begin{align*}v_f = v_i + at \quad \text{so} \quad t=\frac{v_f-v_i}{a}\end{align*}
If we calculate the time required for the ball to rise up to its highest point and come to rest, the initial velocity is 70.7 m/s and the final velocity is 0 m/s. Since we have called the upward velocity positive, then the acceleration must be negative or -9.80 m/s2.
\begin{align*}t=\frac{v_f-v_i}{a}=\frac{0 \ \text{m/s}-70.7 \ \text{m/s}}{-9.80 \ \text{m/s}^2}=7.21 \ \text{s}\end{align*}
Since this is the time required for the cannon ball to rise up to its highest point and come to rest, then the time required for the entire trip up and down would be double this value, or 14.42 s. The maximum height reached can be calculated by multiplying the time for the upward trip by the average vertical velocity. Since the object's velocity at the top is 0 m/s, the average upward velocity during the trip up is one-half the initial velocity.
\begin{align*}v_{\text{up}-\text{ave}} = \left(\frac{1}{2}\right)(70.7 \ \text{m/s}) = 35.3 \ \text{m/s}\end{align*}
\begin{align*}\text{height} = (v_{\text{up}-\text{ave}})(t_{\text{up}}) = (35.3 \ \text{m/s})(7.21 \ \text{s}) = 255 \ \text{m}\end{align*}
The horizontal distance traveled during the flight is calculated by multiplying the total time by the constant horizontal velocity.
\begin{align*}d_{\text{horizontal}} = (14.42 \ \text{s})(70.7 \ \text{m/s}) = 1020 \ \text{m}\end{align*}
Example Problem: A golf ball was hit into the air with an initial velocity of 4.47 m/s at an angle of 66° above
the horizontal. How high did the ball go and how far did it fly horizontally?
Solution:
\begin{align*}v_{\text{i}-\text{up}} = (4.47 \ \text{m/s})(\sin 66^\circ) = (4.47 \ \text{m/s})(0.913) = 4.08 \ \text{m/s}\end{align*}
\begin{align*}v_{\text{i}-\text{hor}} = (4.47 \ \text{m/s})(\cos 66^\circ) = (4.47 \ \text{m/s})(0.407) = 1.82 \ \text{m/s}\end{align*}
\begin{align*}t_{\text{up}}=\frac{v_f-v_i}{a}=\frac{0 \ \text{m/s} - 4.08 \ \text{m/s}}{-9.80 \ \text{m/s}^2}=0.416 \ \text{s}\end{align*}
\begin{align*}v_{\text{up}-\text{ave}} = \left(\frac{1}{2}\right)(4.08 \ \text{m/s}) = 2.04 \ \text{m/s}\end{align*}
\begin{align*}\text{height} = (v_{\text{up}-\text{ave}})(t_{\text{up}}) = (2.04 \ \text{m/s})(0.416 \ \text{s}) = 0.849 \ \text{m}\end{align*}
\begin{align*}t_{\text{total trip}} = (2)(0.416 \ \text{s}) = 0.832 \ \text{s}\end{align*}
\begin{align*}d_{\text{horizontal}} = (0.832 \ \text{s})(1.82 \ \text{m/s}) = 1.51 \ \text{m}\end{align*}
Example Problem: Suppose a cannon ball is fired downward from a 50.0 m high cliff at an angle of 45° with an initial velocity of 80.0 m/s. How far horizontally will it land from the base of the cliff?
Solution: In this case, the initial vertical velocity is downward and the acceleration due to gravity will increase this downward velocity.
\begin{align*}v_{\text{i}-\text{down}} = (80.0 \ \text{m/s})(\sin 45^\circ) = (80.0 \ \text{m/s})(0.707) = 56.6 \ \text{m/s}\end{align*}
\begin{align*}v_{\text{i}-\text{hor}} = (80.0 \ \text{m/s})(\cos 45^\circ) = (80.0 \ \text{m/s})(0.707) = 56.6 \ \text{m/s}\end{align*}
\begin{align*}d = v_{\text{i}-\text{down}} t + \frac{1}{2} at^2\end{align*}
\begin{align*}50.0 = 56.6t + 4.9t^2\end{align*}
Changing to standard quadratic form yields \begin{align*}4.9t^2 + 56.6t - 50.0 = 0\end{align*}
This equation can be solved with the quadratic formula. The quadratic formula will produce two possible solutions for \begin{align*}t\end{align*}:
\begin{align*}t=\frac{-b+\sqrt{b^2-4ac}}{2a}\end{align*} and \begin{align*}t=\frac{-b-\sqrt{b^2-4ac}}{2a}\end{align*}
\begin{align*}t=\frac{-56.6+\sqrt{(56.6)^2-(4)(4.9)(-50)}}{(2)(4.9)}=0.816 \ \text{s}\end{align*}
The other solution to the quadratic formula is -12.375 s. Clearly, the cannon ball doesn't take -12 seconds to fly. Therefore, we take the positive answer. Using the quadratic formula will give you two answers; be careful to think about the answer you get - does it make sense?
To solve the problem, we plug the speed and time into the equation for distance:
\begin{align*}d_{\text{horizontal}} = (0.816 \ \text{s})(56.6 \ \text{m/s}) = 46.2 \ \text{m}\end{align*}
#### Summary
• To calculate projectile motion at an angle, first resolve the initial velocity into its horizontal and vertical components.
• Analysis of projectile motion involves dealing with two motions independently.
• Vertical components will always have the acceleration of gravity acting on them.
• Vertical motion is symmetrical - the distance and time are the same in the rise as in the fall; the final velocity will have the same magnitude as the initial velocity but in the opposite direction.
• Horizontal components will never be effected by gravity; it is constant velocity motion.
#### Practice
The following video shows the famous "shoot the monkey" demonstration. The idea is that if a cannon aimed at a monkey, and fired at exactly the moment the monkey drops, the monkey will always be hit by the cannon ball. The results of the demonstration may be surprising given that it seems that the projectile will miss the monkey because the monkey will fall under the cannon ball.
1. What is the cannon ball in this video?
2. What is used as the monkey in this video?
3. Does the velocity of the cannon ball matter, or will it hit the monkey at any velocity?
#### Review
1. A player kicks a football from ground level with a velocity of magnitude 27.0 m/s at an angle of 30.0° above the horizontal.
1. Find the time the ball is in the air.
2. Find the maximum height of the ball.
3. Find the horizontal distance the ball travels.
2. A person standing on top of a 30.0 m high building throws a ball with an initial velocity of 20. m/s at an angle of 20.0° below horizontal. How far from the base of the building will the ball land?
3. An arrow is fired downward at an angle of 45 degrees from the top of a 200 m cliff with a velocity of 60.0 m/s.
1. How long will it take the arrow to hit the ground?
2. How far from the base of the cliff will the arrow land?
### Notes/Highlights Having trouble? Report an issue.
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Epithelial cells and Characteristics:
Epithelial cells are a uniform cell type, that form the epithelium in organisms. Being stationary, the cells are tightly packed in order to be anchored into the basement membrane. The epithelial cells cover the outside body surfaces, and line any hollow organs within the body, such as respiratory, urogenital and digestive system.
The tissue also forms the lining of body cavities, and can also form glands. Being avascular, epithelial cells do not have blood vessels, and also possess the ability of regeneration through the process of cell division in order to replace any dead cells.
The epithelium is classified by the cell shape, or layer number, of a given cell. When classifying on layer number, three epithelium types are identified – stratified, pseudostratified and simple.
In a simple epithelium, an extension of epithelial cells away from the basement membrane, forms a single layer. When multiple epithelial cell layers are arranged within the epithelium, it is called a stratified epithelium. A pseudostratified epithelium looks to have multiple cell layers, but upon closer inspection, the epithelium is actually connected and attached to the basement membrane.
Mesenchymal cells and Characteristics:
Mesenchymal cells, are a particular group of cells that have similar function and morphology. They make up mesenchymal tissue, which is connective tissue from the three gastrula germ layers. With the ability to differentiate in a variety of different mature cell types, Mesenchymal stem cells are considered multipotent stem cells.
These mesenchymal stem cells differentiate into cells that are required to make connective tissues, adipose tissue, cartilage, bone tissues and lymphatic tissues in adulthood. They are either stellate or fusiform, located between the endoderm and ectoderm of a young embryo, existing in the mesoderm area. The majority of mesenchymal cells will originate in the mesoderm.
The mesenchyme emerges first during gastrulation, as a result of a transitional process known as the mesenchymal-epithelial transition. This basic process occurs during regeneration of the tissue from the embryo. Embryonic mesenchymal cells can become epithelial cells, as can epithelial cells become mesenchymal, with the transition process being reversible. A conversion of epithelial into mesenchymal cells is initiated through a loss of epithelial, adherens junctions, tight junctions and cadherin on epithelial cell membranes.
Cell surface molecules on the epithelial cells, will undergo endocytosis, loosening the shape of the microtubule cytoskeleton, which enables mesenchymal cells the ability of migration along the extracellular matrix. Whenever secondary generation of epithelial tissue is required, a conversion of mesenchymal cells into epithelial cells will occur, showing the reverse transitional process.
Difference between Epithelial and Mesenchymal cells
The key differences between epithelial and mesenchymal cells, is the difference in cell type and the way they differentiate.
Epithelial cells are a uniform cell type, making up the epithelium in the body tissues. Mesenchymal on the other hand, are a multipotent cell type, originating in the mesoderm.
When they differentiate, epithelial cells change in order to cover surfaces on the body and line body cavities and hollow organs. Mesenchymal, differentiate into cells that are used to create connective tissue, adipose tissue, cartilage, bone tissues and lymphatic tissues.
Table of comparison to show the difference between Epithelial and Mesenchymal cells:
Both epithelial and mesenchymal cells are differentiated vertebrate cell types. Epithelial cells are uniform and adhere tightly to one another to form the epithelium tissue. This epithelium is a protective layer that covers body cavities and surfaces.
Mesenchymal cells are a multipotent cell type, and originate in the mesoderm. The Mesenchymal stem cells, can differentiate into a variety of different cell types, and change into cells that are required in order to create connective tissues, adipose tissue, cartilage, bone tissue and lymphatic tissues in adulthood. This represents the key difference between the two types of cells – the ability to differentiate into different cells.
Author: Alex Hammond
Alexander Hammond hold a first-class master’s degree in Ecology. He has conducted a number of international research projects in Indonesia, Belize and the UK, in the areas of Marine Biology, Terrestrial Ecology and Conservation. Several of his research reports have been published.<|endoftext|>
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0
# What is the answer when you multiply 2 with 9?
Updated: 10/23/2022
Wiki User
11y ago
18. Let me show you why:
If I have 1 set of 9, I have 9
. . . . . . . . .
2 sets of 9 makes 18
. . . . . . . . . . . . . . . . . .
Wiki User
11y ago
Earn +20 pts
Q: What is the answer when you multiply 2 with 9?
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Related questions
51
### What are all the ways you can multiply and get 18?
You can multiply 2 and 9 to get 18. Alternatively, you can multiply 3 and 6 to get 18 as well.
1/2 x 9
### What is a 3 digit number divisible by 2 and 9?
If you multiply 2*9, you get 18. Multiply that by 10 to get a three digit number, and you get 180.
### what is 2|9 of 36 in fraction?
2/9 of 36= 8 To find the answer to this you should divide 36 by the denominator and multiply by the numerator. So: 36 divided by 9 and then multiply by 2 36/9= 4 4*2= 8
### What two numbers can multiply to get 9?
How about: 2*4.5 = 9 or 4*2.25 = 9
63
### How do you get from 1.8 to 2?
Multiply by 10/9 = 1.111...
1-9
### How do you check a division problem?
Multiply the answer to the problem by the number that you are dividing by. (e.g. 18/2=9 Check it with 9*2=18)
### What is four and a half multiplied by two and one ninth in simplest form?
To multiply fractions, you multiply the numerators (4.5 * 2 = 9) and multiply the denominators (1 * 9 = 9). Thus, four and a half multiplied by two and one ninth in simplest form is 9/9, which simplifies to 1.
### How do you evaluate 2xy times 9 if x equals 3 and y equals 2?
2xy*9 if x=3 and y = 2 2(3)(2)*9 12*9 108 You substitute x and y for their equivalent (3 and 2) then simply multiply 2*x*y, then multiply the answer of that by 9.<|endoftext|>
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The Jacksons also acted as guardians for eight other children. Lawrence pulled out a second pistol which also misfired. In the mid-1850s second-hand evidence indicated that he may have been born at a different uncle's home in North Carolina.
He strengthened the power of the presidency which he saw as spokesman for the entire population as opposed to Congressmen from a specific small district. Jackson was nicknamed "Old Hickory" because of his toughness and aggressive personality; he fought in duels some fatal to his opponents. Whigs and moralists denounced his aggressive enforcement of the Indian Removal Act which resulted in the forced relocation of thousands of Native Americans to Indian Territory (now Oklahoma).<|endoftext|>
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Granulocytes are a group of immune cells that are important in fighting pathogens and healing damaged cells. They play critical roles in inflammation and wound healing. High levels occur in infections, autoimmune diseases, and many cancers. Low levels are most often caused by infections and drugs. Read on to find out what high and low granulocyte levels mean and how you can decrease or increase your levels.
What Are Granulocytes?
Granulocytes are a type of white blood cell that contains small sacs called granules. The contents of these granules are released into the blood during infections, injuries, and allergic reactions. These contents include antimicrobial proteins, enzymes to digest bacteria, and reactive oxygen species [R].
There are four types of granulocytes [R]:
Granulocytes are made in the bone marrow from stem cells and then released into the circulation. They are part of the innate immune system, which provides a quick, general response to pathogens. Granulocytes become activated by pathogens and damaged cells .
Granulocytes are also known as polymorphonuclear leukocytes (PML) or polymorphonuclear neutrophils (PMN).
The main function of granulocytes is to engulf and destroy invading pathogens and parasites. They are responsible for starting the process of inflammation as well as resolving it. Granulocytes are also involved in wound healing and tissue remodeling [4, 5].
Immune messenger molecules (eotaxin and IL-8) recruit granulocytes from the bloodstream to injured or infected tissues. They are then activated by bacteria, viruses, and fungi or damaged cells. When they encounter a pathogen, granulocytes engulf them. They then release the contents of their granules to digest and destroy them. However, they also can cause damage to your own cells in the process .
After a threat has been eliminated, granulocytes destroy themselves by programmed cell death (apoptosis). However, in many inflammatory diseases like rheumatoid arthritis and asthma, granulocytes last longer than they should .
Neutrophils are the most abundant granulocytes as well as the most abundant immune cells in the body. They make up 50% to 70% of all immune cells. They are very short-lived and only survive 8-12 hours in the blood (where they normally reside) and 1-2 days in tissues (when there is an infection) [2, 6, 6].
Neutrophils are among the first immune cells to arrive at the site of an injury or infection. They produce antimicrobial compounds called defensins, enzymes (proteases), and reactive oxygen species (superoxide and hydrogen peroxide) that break down and destroy pathogens. However, these compounds can also cause damage to the surrounding tissue. This can lead to delayed healing and excess scar tissue formation [7, 7].
Eosinophils were thought to primarily fight multicellular parasites such as worms. However, recent research suggests that they may also allow some parasites to live while preventing them from doing too much damage. Together with mast cells and basophils, eosinophils play important roles in the development of allergy and asthma. They also play many other important roles including [9, 10, 11, 12, 13]:
- Fighting viral, bacterial, and fungal infections
- Activating acquired immunity (part of the immune system that develops a “memory” for pathogens)
- Helping prepare the uterus for pregnancy
- Repairing and remodeling tissue
- Regulating blood sugar and insulin levels
Like neutrophils, they are made in and mature in the bone marrow and then released into the bloodstream. However, unlike neutrophils, they are only in the blood for a short time (9-18 hours). Instead, they take up residence in the gut, ovaries, and lymph nodes, where they can live for several weeks. They are usually not found in the lung, skin, or throat except in disease states [11, 14, 15].
Basophils help fight bacteria, viruses, and parasites. They also play key roles in allergies and autoimmune diseases. They are the largest yet least common granulocyte in the body. Basophils reside in the bloodstream and are recruited to sites of inflammation when needed [16, 17].
When basophils become activated, they release the contents of their granules, which includes heparin and histamine. Histamine expands blood vessels and increases blood flow. Heparin is an anti-clotting agent that helps maintain proper blood flow. This allows immune cells easy access to the site of inflammation [20, 21, 22].
Due to their similarity, mast cells were originally thought to be basophils. Whereas basophils mainly circulate in the bloodstream, mast cells live in most tissues throughout the body. They are especially abundant in places that come into close contact with the environment, such as the skin, the gut, and the airways [17, 23, 24].
Because of their location, mast cells are responsible for the early recognition of foreign invaders. Within seconds of encountering a pathogen, mast cells release histamine, enzymes, and heparin .
Mast cells can contribute to wound healing and the growth of new blood vessels (angiogenesis) .
Mast cells also play key roles in asthma and allergies by overreacting to the presence of harmless substances such as pollen and pet dander. They are the main contributors to the symptoms associated with these conditions. Mast cells also play roles in autoimmune conditions, including rheumatoid arthritis and multiple sclerosis .
By releasing pro-inflammatory cytokines (TNF-α and IL-6), mast cells can recruit T cells and dendritic cells to help fight pathogens. This makes mast cells a crucial link between the innate and adaptive immune systems .
Immature granulocytes are granulocytes that are still maturing. They are normally located in the bone marrow and have not yet matured into granulocytes [26, R].
Granulocytes Normal Range
Granulocytes are measured as part of a standard comprehensive blood count (CBC) test.
The normal range of granulocytes is 1.5 – 8.5 x 10^9/L or between 1,500 and 8,500 cells per microliter (mL) of blood.
Levels below this range are referred to as granulopenia. Granulopenia is commonly referred to as neutropenia (low neutrophil levels). Severely low levels (below 500 cells/µL) are referred to as agranulocytosis. Low levels of granulocytes reduce the body’s ability to fight infections .
Levels above this range are referred to as granulocytosis. Granulocytosis is commonly referred to as neutrophilia.
Causes of High Granulocytes
The sympathetic nervous system is responsible for the body’s fight-or-flight response. Granulocyte counts are normally elevated during pregnancy due to a higher sympathetic nervous system activity. This increased activity helps the developing fetus get enough oxygen and nutrients [30+].
2) Intense, Prolonged Exercise
Several studies of 141 total people have found that intense exercise including endurance exercise and strength training can substantially increase neutrophil levels. Exercise causes neutrophils to be released from the bone marrow at a higher rate [31, 32, 33, 34, 35, 36, 37, 38].
Smoking was strongly linked to high granulocytes in a study of 38k people. In a study of 1,730 people, granulocytes levels decreased after smoking was stopped .
4) Sleep Loss
Sleep is incredibly important to the function of your immune system. In a pilot of study eight people, sleeping only four hours a night for three nights increased neutrophil levels by 34% .
Neutrophil levels increased by 30% after a single night of no sleep in 16 people (RCT) .
6) Heavy Metal and Chemical Poisoning
- Insect venom
7) Cushing’s Syndrome
Granulocyte levels increased in ten patients who underwent elective spine surgery due to an increase in cortisol .
Granulocyte levels were substantially higher in patients with appendicitis in a study of 456 people .
By increasing a protein called granulocyte colony stimulating factor (G-CSF), many cancers can cause the body to make too many granulocytes. Cancers that result in high granulocyte levels include [48, 49, 50, 51, 52]:
- Lymphoma (cancer of the lymphatic system)
- Chronic myeloid leukemia (cancer of the bone marrow)
11) Heart Attack
Neutrophils play an important role in repairing the damage done to the heart after a heart attack. Levels will greatly increase in the hours following a heart attack and are directly related to the degree of damage [53, 54].
Diseases and Conditions Associated With High Granulocytes
1) High Blood Pressure
In a 40-year observational study of 9.4k people, high neutrophil levels increased the risk of developing high blood pressure .
Higher neutrophil levels were linked to fatigue due to stress from work in an observational study of 213 people .
3) Heart Disease (likely causal)
Neutrophils can accumulate in plaques in the artery walls and contribute to hardened arteries (atherosclerosis) .
High neutrophil and eosinophil levels are linked to an increased risk of heart disease. High neutrophils levels are also linked to increased risk for heart attack, stroke, and death from heart disease [58, 59, 60, 61, 62, 63, 64, 65].
Ways to Decrease Granulocytes Levels
1) Vegan and Ketogenic Diets Decrease Granulocytes
2) Fasting Decreases Granulocytes
Ramadan is a religious practice in Islam that involves a month of intermittent fasts for most of the day (sunrise to sunset).
In a study of 28 Muslims, Ramadan decreased neutrophils by 18% .
Another study found Ramadan reduced neutrophils by 7% in 90 Muslims .
While fasting decreases neutrophils, it also improves their ability to engulf and destroy pathogens .
3) Getting Enough Sleep Prevents Increases in Granulocytes
4) Boswellia serrata Decreases Granulocytes
Boswellia serrata gum resin decreased eosinophils in a study 80 people (DB-RCT)
5) Garlic (Allicin) Decreases Granulocytes
Neutrophil levels increased in rabbits infected with P. multocida bacteria. Allicin, the main active component of garlic, decreased neutrophil levels .
6) Reducing Stress May Prevent Increases in Granulocytes
Stress may increase neutrophil levels, which means that avoiding or managing stress may help prevent your neutrophil levels from increasing .
Causes of Low Granulocytes
1) Vitamin B9, Vitamin B12, and Iron Deficiencies
Iron deficiency can also lead to granulopenia. However, the mechanism for this is still unknown .
Certain bacterial, viral, protozoan and fungal infections can cause low granulocyte levels. The flu, Epstein-Barr virus (EBV), cytomegalovirus (CMV), and hepatitis A, B and C are common viruses that can lead to low granulocyte levels [43, 80, 78, 79].
3) Autoimmune Disease
Systemic lupus erythematosus, also known as lupus, is an autoimmune disease. Neutrophils die at a much quicker rate in people with lupus. Because of this, low granulocyte levels are also seen in people with lupus, with 50% having abnormally low levels [83, 84, 77, 82].
Low granulocyte levels are also a common feature of Sjögren’s syndrome, an autoimmune disease that causes dry eyes and mouth .
4) Bone Marrow Disorders
Bone marrow disorders can decrease granulocytes by interfering with their production. Examples of bone marrow disorders include :
- Leukemia (bone marrow cancer)
- Aplastic anemia
6) Inborn Neutropenia
Congenital (inborn) neutropenia is a condition of low neutrophil levels from birth due to genetic disorders .
One example of congenital neutropenia is a condition called benign ethnic neutropenia (BEN). BEN is caused by a small genetic mutation. It is found in 25% to 50% of people of African descent and some ethnic groups in the Middle East. However, it is not associated with an increased risk of infection that is commonly seen in other forms of congenital neutropenia [90, 91].
Other forms of congenital neutropenia include Kostmann’s syndrome and cyclic neutropenia .
7) Enlarged Spleen
8) Organ Transplants
Low neutrophil levels occur in up to 28% of kidney transplant and 24% of liver transplant recipients during the first year. Low neutrophil levels are associated with more infections and organ rejection in kidney transplant recipients, and bloodstream infections and increased risk of dying in liver transplant recipients .
9) Thyroid Disorders
Thyroid disorders (hypothyroidism and hyperthyroidism) are found in up to 43% of people with granulopenia. Both low and high thyroid hormones are thought to cause the destruction of granulocyte precursors. People with thyroid disorders also have high levels of antibodies to granulocytes, which causes the body to destroy them [94, 95].
Hemodialysis is the use of a filter to clean the blood of people whose kidneys are not working properly. The process changes neutrophils so that they get stuck in the blood vessels, leading to low levels in the bloodstream [96, 97].
11) Severe Burns
Diseases and Conditions Associated With Low Granulocytes
2) Type 1 Diabetes
Type 1 diabetes is an autoimmune disorder that causes the destruction of the cells that produce insulin (beta cells). Low neutrophil levels were associated with an increased risk of developing type 1 diabetes in a study of 436 people .
3) Obsessive-compulsive Disorder
Ways to Increase Granulocytes Levels
1) Intense Exercise Increases Granulocytes
2) Dark Chocolate Increases Granulocytes
3) Caffeine Increases Granulocytes
Caffeine increased both neutrophils and eosinophils in mice .
In a study (DB-RCT) of 22 people, caffeine increased neutrophil levels by 9%. In the group that exercised and supplemented with caffeine, neutrophil levels were increased by 58% .
Effects of Drugs on Granulocytes Levels
Drugs that increase granulocyte levels include:
Drugs that decrease granulocyte levels include:
- ACE inhibitors [110, 111]
- Allopurinol
- Pain-relieving drugs
- Antibiotics
- Anticoagulants
- Antidepressants
- Antidiabetics
- Antiepileptics [115, 116, 117]
- Antihistamines
- Antimalarial
- Antipsychotics [118, 119]
- Antithyroid medication [120, 121, 122]
- Chemotherapy [123, 124]
- Cimetidine
- Cocaine
- Colchicine
- Dapsone
- Diuretics
- Griseofulvin
- Immunosuppressives
- Levodopa
- Levamisole
- NSAIDs
Genetics of Granulocytes
Gene Mutations and Neutropenia Risk
VPS45 is a gene that encodes a protein that helps control membrane trafficking. In a study of 7 children with mutations in the gene, the children had low neutrophil levels (neutropenia) and neutrophil dysfunction .
Mutations in the CXCR4 gene are associated with WHIM syndrome, a rare disease where the body’s immune system does not function properly. WHIM patients have severe low neutrophil levels (neutropenia) because neutrophils do not exit the bone marrow .
People with Kostmann’s disease have neutrophil levels lower than 0.2 ×109/l. Some of the patients have mutations in ELA2 or HAX-1. Additionally, Kostmann’s patients can also acquire CSF3R gene mutations .
Chediak-Higashi patients have CHS1 gene mutations, which can cause neutropenia .
Shwachman-Diamond is a rare disorder, where the patients have defective neutrophil movement in the blood. The SBDS gene has a mutation that is associated with neutropenia .
Gene Mutations and Neutrophilia Risk
A mutation in the CSF3R gene promotes neutrophil formation. This can lead to higher susceptibility for hereditary chronic neutrophilia .
PSTPIP1 mutation can play a role in neutrophilic dermatoses susceptibility .
People with the GPSM3 SNP rs204989 may have decreased GPSM3 production and be protected against rheumatoid arthritis. This specific variation reduces neutrophil movement to the inflammation site, which prevents long-term inflammation that is associated with arthritis .
A mutation in the RAC2 gene is associated with neutrophil dysfunction and can cause a person to be predisposed to bacterial infections. It is also associated with human immunodeficiency syndrome .
Pelger-Huët anomaly is a genetic disorder where the nucleus of neutrophils is in odd shapes. However, patients with this disorder are mostly healthy and neutrophils still function normally. A mutation in the LBR gene causes Pelger-Huët anomaly .
Eosinophils and Basophils
Irregular Granulocyte Levels?
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They form a circle, and following introductions, the teacher creates a movement-sound sequence figuratively faithful to motives from Mahler’s Symphony No 5 first movement, the ‘Trauermarsch’.
The musical material transmitted is Mahler’s. There are 15 minutes of intensive working where the teacher gives and the pupils give back, where the teacher insists through repetition that all get it. The transaction is already playful and relational. Like catching balls moving fast between all within the circle, the pupils catch melodic fragments as well as rhythmic ones.
‘You really need to get hold of this material, this is very important’, says the teacher.
Now with a voice of enchantment and mystery the teacher reveals Mahler’s use of the song ‘Der Tambourg’sell’, a song about one of Mahler’s ill-fated ‘children’, a drummer boy condemned to execution and his long walk to death, the ‘trauermarsch’.
The pupils want to know what it is that the boy has done that deserves such a fate. However, this is to remain a mystery for the time being. The work proceeds until groups have created their own ‘trauermarschen’ using Mahler’s material.
In the minds of the pupils live the drummer boy and his fate and the musical ideas and feelings that in some sense are now theirs as well as Mahler’s. The pupils remain curious, continually asking questions of their teacher and each other.
- Why do teachers ask questions?
- Why are children expected to compose music without first experiencing a felt provocation to do so?
- Do such provocations lead to composing music that has stronger character and thicker meanings?
- Why does much music education have so little human interest?
- Why do music teachers teach musical skills without rich content?<|endoftext|>
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Vibrant but toxic, poison arrow frogs range from less than an inch to two and a half inches in body length. There are more than 100 species of poison dart frogs, varying in color and pattern. The black and green species has black spots, the strawberry or blue jeans frog is all red with blue legs, the yellow-banded species appears painted with yellow and black. Color shades vary among frogs within a species. It is the skin that contains the frog's poison.
These beautiful colors are warnings to potential predators that the frogs are poisonous. Other species, such as monarch butterflies, sport bright colors to advertise their toxicity. Several species of non-poisonous frogs evolved with similar coloring to avoid being eaten. Some scientists think that the reticulated pattern of the frogs also acts as camouflage among the forest shadows.
Distribution and Habitat
Poison dart frogs live in the rainforests of Central and South America and on a few Hawaiian islands.
Poison dart frogs feed mostly on spiders and small insects such as ants and termites, which they find on the forest floor using their excellent vision. They capture their prey by using their long sticky tongues.
Male frogs go through an elaborate ritual to attract a mate. The males vocalize, a loud trill sound, to attract females. Once the courtship ritual is complete, the females deposit dozens of eggs on leaves. The eggs are encased in a gelatinous substance for protection against decay.
During the two-week development period, the male returns to the eggs periodically to check on them. Once the tadpoles hatch, they swim onto the male’s back and are attached by a mucus secretion, which keeps them from falling off. The male carries them to a place suitable for further development, such as wet holes in broken trees and branches, little ponds, wet coconut shells, and even in tin cans and car tires. Tiny pools of water that collect in bromeliads are also used by some species.
Once at their final destination, the tadpoles are on their own. They need an additional three months to metamorphose into small frogs.
They may live more than ten years in captivity.
Some poison dart frogs are endangered due to habitat loss, which is causing numbers to decline among many species.
The possibility of new medications from these frogs' secretions is being explored.
Poison dart frogs, also called poison arrow frogs, are so named because some Amerindian tribes have used their secretions to poison their darts. Not all arrow frogs are deadly, and only three species are very dangerous to humans. The most deadly species to humans is the golden poison arrow frog (Phyllobates terribilis). Its poison, batrachotoxin, can kill many small animals or humans. These frogs are found in Colombia along the western slopes of the Andes. Arrow frogs are not poisonous in captivity. Scientists believe that these frogs gain their poison from a specific arthropod and other insects that they eat in the wild. These insects most likely acquire the poison from their plant diet.<|endoftext|>
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# How do you divide fractions with different exponents?
## How do you divide fractions with different exponents?
In order to divide exponents with different bases and the same powers, we apply the ‘Power of Quotient Property’ which is, am ÷ bm = (a ÷ b)m. For example, let us divide, 143 ÷ 23 = (14 ÷ 2)3 = 73.
What are the rules for multiplying and dividing exponents?
There are different rules to follow when multiplying exponents and when dividing exponents. If we are multiplying similar bases, we simply add the exponents. If we are dividing, we simply subtract the exponents. If an exponent is outside the parentheses, it is distributed to the inside terms.
What do you do with exponents when multiplying?
When you’re multiplying exponents, use the first rule: add powers together when multiplying like bases. 5^2 × 5^6 =? The bases of the equation stay the same, and the values of the exponents get added together.
### Do you add or multiply exponents when dividing?
To divide exponents (or powers) with the same base, subtract the exponents. Division is the opposite of multiplication, so it makes sense that because you add exponents when multiplying numbers with the same base, you subtract the exponents when dividing numbers with the same base.
How do you multiply with exponents?
Multiplying exponents with different bases First, multiply the bases together. Then, add the exponent. Instead of adding the two exponents together, keep it the same. This is because of the fourth exponent rule: distribute power to each base when raising several variables by a power.
Do you multiply exponents when multiplying?
#### Do you distribute or exponents first?
Explanation: First simplify the expression inside the parentheses. Then distribute the exponent. Rearrange the expression so that there are no more negative exponents.
What are the rules for multiplying exponents?
When multiplying exponential expressions that have the same base, add the exponents. When dividing exponential expressions that have the same base, subtract the exponents. When raising an exponential expression to a new power, multiply the exponents.
Do you add exponents when multiplying?
The exponent “product rule” tells us that, when multiplying two powers that have the same base, you can add the exponents.
## What are the 5 rules of exponents?
Conclusion: exponent rules practice
• Product of powers rule — Add powers together when multiplying like bases.
• Quotient of powers rule — Subtract powers when dividing like bases.
• Power of powers rule — Multiply powers together when raising a power by another exponent.
• August 28, 2022<|endoftext|>
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Esotropia can be divided into various categories each requiring a different treatment plan; each having a different prognosis.
- Congenital Esotropia
- Infantile Esotropia
- Esotropia with Amblyopia
- Accommodative Esotropia
- Partially Accommodative Esotropia
“Congenital” means from birth and, using this strict definition, most infants are born with eyes that are not aligned at birth. Only 23% of infants are born with straight eyes. In the majority of cases, one eye or the other actually turns outward during the neonatal period. Within the first three months the eyes gradually come into more consistent alignment as coordination of the two eyes together as a team develops.
It is common for infants to appear as if they have esotropia, or inward turn of the eyes, because the bridge of the nose is not fully developed. This false or simulated appearance of an inward turning is known as epicanthus. As the infant grows, and the bridge narrows so that more of the white of the eyes (sclera) is visible on the inner side, the eyes will appear more normal.
True congenital esotropia is an inward turn of a large amount, and is present in very few children, but the infant will not grow out of this turn. True infantile esotropia usually appears between the ages of 2 and 4 months.
The baby with infantile esotropia usually cross fixates, which means that he or she uses either eye to look in the opposite direction. The right eye is used to look toward the left side, and the left eye is used to look toward the right side. By definition, they alternate which eye they are looking with. It is more difficult to help this type of strabismus with non-surgical methods, such as Vision Therapy and/or glasses. Sometimes, clear tape applied to the inner third of each lens (occlusion) can reduce the tendency to turn inward. Prisms may aid alignment if the turn is not too large.
Some children who develop strabismus, in which coordination between the two eyes is poor, also have atypical gross motor development patterns. They typically skip the crawling stage with bilateral movements, and go right from creeping to standing. The interplay between gross motor, particularly balance systems (cerebellar and vestibular) and binocular systems (motor control of the two eyes) is also evident in the large number of young children with cerebral palsy who have strabismus.
If the inward turn of the eye is constant, and of a large amount, surgery may be recommended by some health care professionals but many times multiple procedures are needed to obtain perfect alignment of the two eyes for the patient. Furthermore, even multiple surgeries or “revisions” may end up yielding cosmetic benefits only. That is, the two eyes might look normal or “straight” to outside observers, but normal two-eyed vision has not been achieved.
Improvement might only be cosmetic as surgery does not necessarily enable the brain to utilize information from both eyes simultaneously (binocular vision), so eye teaming, eye tracking, stereoptic vision and/or 3D depth perception is often poor following surgical treatment. If surgery is undertaken, the best chance for visual success occurs when the surgeon works with a developmental optometrist who is comfortable in prescribing glasses and Optometric Vision Therapy to encourage perfect alignment of the two eyes with proper fusion and eye teaming. Such a model of cooperative care would be similar to the complementary relationship between an orthopedic surgeon and a physical therapist.
If Amblyopia is present, therapy including binasal medial occlusion, correct eyeglass prescription, prisms, and/or optometric vision therapy is often required so that the turned “lazy” eye develops the capacity to see as well as the preferred eye. This promotes binocular vision (using both eyes together) and provides the greatest outcome.
If excessive inward turning of an eye is first noted around 2 years of age, it may be due to difficulty integrating the focusing (accommodative) system with the eye alignment (binocular) system. Normally when we look across the room or beyond, our eyes are parallel, or straight. However, when we look at things up close, two things happen. We need to converge more (aim both eyes inward at the same time) and we have to input more focus, or accommodate to keep things clear. Children have large amounts of focusing power, and sometimes in getting things clear, inward turning or esotropia results. If the inward turning only occurs up close, as when playing with small objects, making eye contact, coloring, looking at picture books and so forth, the child may just need glasses for near activities to reduce or eliminate the esotropia.
However, if a child is significantly farsighted (hyperopia), an inward turn of the eye may even occur when focusing to look further away, such as television. If the amount of turn is greater at near than far, your optometrist may prescribe a multifocal lens. For children this could be a traditional bifocal with a line, or a form of no-line bifocal or progressive lens. Your optometrist will review with you which is the best option for your child. In addition, Vision Therapy may be of benefit. Accommodative Esotropia should never be treated with surgery.
When the eyes are aligned by corrective lenses sometimes the eyes spontaneously begin to work together. Other times, they need help. Remember, the habit of suppressing or turning off one eye or the other was probably developed over a number of years. The eyes have to be trained to work together again and suppression must be eliminated in order to restore normal eye teaming, depth perception, and stereopsis. The eye doctor might have to patch an eye that was suppressed or turned off and/or employ Vision Therapy.
Intermittent turns usually do not require long term treatment. Vision Therapy may be necessary to improve the muscle coordination and eventually eliminate the bifocal.
Patients with Accommodative Esotropia should never have eye muscle surgery to eliminate the need for glasses. If they do, they will have significant focusing problems when they get older.
In some instances, part of the inward turn is due to basic esotropia, and an additional amount due to the effect of accommodation. Glasses may reduce the amount of eye turn, but it is not totally compensated. Initially, the eye doctor may prescribe prism to compensate for the amount of turn. Office-based Vision Therapy is usually needed. Because vision is a learned process, therapy is often helpful in learning new binocular vision patterns, or restoring normal pathways that have been lost or underutilized. Binocular vision occurs in the visual centers of the brain, not in the eye muscles.<|endoftext|>
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Joyce Clements Artist PO Box 39 Bolinas CA 94924 415-868-1008
Life begins for the Leatherback turtle when it cracks through the shell of an egg and makes its way out. The hatchling has from sundown to sunrise to dig its way out of the sand nest and find its way to the ocean. If a baby does not reach the sea by daybreak it risks death by dehydration or being eaten by a predator.* Under “normal” circum-stances, only about 1 in a 1,000 of the baby turtles will survive the 15 years it will take to reach maturity.
Leatherback turtles have existed since dinosaurs roamed the planet some 100
million years ago. By 1982, there were an estimated 115,000 reproductive female leatherbacks left in the world. Now, 24 years later, there are fewer than 3,000 of them alive, a decline of 97%.
Once newly hatched turtles enter the sea, only the females will ever return to land. Mature sea turtles mate at sea, and females return to the beach where they were born to lay clusters of eggs in nests they dig in the sand. Some of the eggs in the cluster are yolkless “dummy eggs” which protect and buffer the fertilized eggs from cracking. The eggs incubate approximately 2 months before the young hatch.
What are we doing that is leading the sea turtles toward extinction? Factors include killing the turtles for their meat, poaching their eggs, capturing and killing them as by-catches in net and long-line fishing, destroying or compromising their nesting and birthing sites through commercial development and tourism, and polluting their environment. Turtles are killed and injured by collisions with ships. They die from consuming floating trash like plastic bags which they mistake for their main food source, jellyfish.
In addition to the Leatherback, five of the six other sea turtle species are also at risk of extinction.<|endoftext|>
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Recent celebrity suicides have shed more media light on the issue of depression and suicide. Any time someone commits suicide, family, friends and acquaintances are left in shock and sadness. There is often an overwhelming question of why and why didn’t we know? These questions are also followed by feelings of guilt and regret for not having known how badly a loved one was suffering.
It can be a difficult and awkward topic for parents to discuss with their children, however it’s important to be direct and as honest as possible when speaking to children about depression and thoughts of self-harm.
Be direct and provide age appropriate information. Provide an explanation of mental illness that makes sense for your child’s age, maturity and level of understanding. For example, for a younger child you may say that ‘people’s thoughts and feeling are controlled by their brain and sometimes their brain gets sick the same way a body can get sick. When someone’s brain gets very sick, it sometimes makes them want to stop their body from working. For an adolescent or teenager, you may use more direct language.
Encourage your child to ask questions. Providing the opportunity to have a conversation about mental health opens the doors to further conversations and it also normalizes discussion about mental health in general. While it may be uncomfortable, try to remain present and listen to your child as much as possible. Practice reflective listening and ask open ended questions such as, ‘How do you feel about what happened?’ ‘What are your thoughts about what happened?’ ‘What questions do you have?’
Talk about the signs and symptoms of depression. If your child is a young adolescent or teenager, it’s a good opportunity to talk about how anxiety and depression affect someone’s behavior. This is the age where kids start sharing less with their parents and more with their peer group, so give your child helpful information so that if they or a friend is feeling depressed, they know what to do.
Finally, emphasize the importance of maintaining good mental health. Just as going to the gym regularly can help keep your body healthy, talking to a licensed therapist or counselor helps keep our minds healthy. Encourage your child to speak and use outlets for their feelings. Let them know who the counselor is in their school, discuss the value of therapy. Consider making an appointment for your child with a therapist if you have any concerns that they may be depressed or anxious.
*If you or your child is feeling suicidal, call 911 or go to your nearest emergency room or contact the National Suicide Prevention Lifeline at 1-800-273-8255 or https://suicidepreventionlifeline.org/. *<|endoftext|>
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What is digital literacy? Looking for ways to build digital literacy in your students by integrating literacy skills and technology into the content areas? A common working definition for digital literacy is:
“Digital literacy is the skills, knowledge, creativity and attitudes that are necessary to be able to use digital media for learning and mastering in the Knowledge Society. It is this literacy that builds a bridge between skills such as reading, writing and arithmetic, and the literacy that is needed to be able to use new digital tools and media in a creative and critical way.”
Digital literacy, when applied to media in print and electronic form, can help students build reading, writing, listening, speaking, and viewing skills. In Elaine Roberts and Debra Coffey’s Keys to Literacy Instruction for the NET Generation, Grades 4-12, the authors compiled a list of web tools to aid students in making sense of the content and move toward a student-centered, project-based learning environment, I’ve included a few here.
Glogster.edu is a presentation tool for students to publish information through images, links, videos, audio, and texts. Most often Glogters surface in classrooms as digital posters, usually to inform on a topic. What would an interactive Glogster look like? Could students collaborate with students inside and outside of their classroom? What sort of viewing tools would make sense for students to use as they read and respond to a glog?
Creaza is a presentation tool for students to create and communicate with their audiences in the form of film, complex text, radio commercials, radio plays, mind minds, presentations, cartoon, digital stories, etc. In this example, the author imported original cartoon drawings into the movie maker to make this short film. How might collaboration inside of a Creaza presentation look inside an elementary, middle, or high classroom? Who might be the audience for these publishable pieces?
Class Dojo is a behavior management system that allows a teacher to enter his/her students and track their behavior or actions with a point system. For instance, a teacher can give a student a point for helping others, being on task, participating, persistence, team work, and/or working hard. If the teacher decides to project the class roster, students have access to immediate feedback to know when they receive points for these behaviors and actions. All collected data can be shared with students, parents, and administrators. What is the added value of a management system such as this?
Game Star Mechanic is a program that allows student to design their own video games and teachers to share lesson designs. The literacy skills the program taps into include 21st century skills, systems thinking, creative problem solving, art and aesthetics, writing and storytelling, and STEM learning. Multiple content specific lesson plans are available online as models. Game Based Learning (GBL), as we’re learning, does have classroom benefits including reinforcing content through technology-enhanced learning experiences. Can game designs accurately target content standards while being relevant to students?
What tools are you using to build digital literacy in your students? What impact are those tools having on the learning experience? Share your practices inside this global community.<|endoftext|>
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The Griffin is a mythical beast with the body of a lion and the head of an eagle. Griffins came into Greek mythology as guardians of the treasures of Apollo. In Christian art, they functioned originally as symbols of Satan but later were used to sumbolize the union of Christ's divine and human natures (Impelluso, 374).
A late example of the satanic symbolism is outside St. Anthony's basilica, where two huge griffins grasp men and beasts in their paws (second picture at right). Perhaps the most notable example of the christological symbolism is in Dante's Purgatorio used the Griffin to represent Christ in the (cantos 29:106-114, 31:118-26).
Griffins also appear in heraldry. The one shown on the right, in a broken fragment from a French monastery, could have been intended either as a symbol of Christ or as a heraldic reference to a sponsor.
Prepared in 2018 by Richard Stracke, Emeritus Professor of English, Augusta University.<|endoftext|>
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Goal 5: Gender equality
Ending all discrimination against women and girls is not only a basic human right, it’s crucial for sustainable future; it’s proven that empowering women and girls helps economic growth and development.
UNDP has made gender equality central to its work and we’ve seen remarkable progress in the past 20 years. There are more girls in school now compared to 15 years ago, and most regions have reached gender parity in primary education.
But although there are more women than ever in the labour market, there are still large inequalities in some regions, with women systematically denied the same work rights as men. Sexual violence and exploitation, the unequal division of unpaid care and domestic work, and discrimination in public office all remain huge barriers. Climate change and disasters continue to have a disproportionate effect on women and children, as do conflict and migration.
It is vital to give women equal rights land and property, sexual and reproductive health, and to technology and the internet. Today there are more women in public office than ever before, but encouraging more women leaders will help achieve greater gender equality.
Women earn only 77 cents for every dollar that men get for the same work.
1 in 3
35 percent of women have experienced physical and/or sexual violence.
Women represent just 13 percent of agricultural landholders.
Almost 750 million women and girls alive today were married before their 18th birthday.
2 of 3
Two thirds of developing countries have achieved gender parity in primary education.
Only 24 percent of national parliamentarians were women as of November 2018, a small increase from 11.3 percent in 1995.<|endoftext|>
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## What are Mixed Numbers?
A mixed number is a entirety number, and also a correct fraction stood for together. It mainly represents a number in between any 2 whole numbers.
You are watching: Another name for a whole number and fraction or decimal
Look at the provided picture, it represents a portion that is higher than 1 yet much less than 2. It is hence, a blended number.
Some other examples of blended numbers are
Parts of a combined number
A mixed number is formed by combining three parts: a whole number, a numerator, and also a denominator. The numerator and also denominator are component of the appropriate fractivity that renders the blended number.
Properties of mixed numbers
It is partially a totality number.
It is partly a fraction.
Converting improper fractions to mixed fractions.
Step 1: Divide the numerator by the denominator.
Step 2: Write dvery own the quotient as the entirety number.
Step 3: Write dvery own the remainder as the numerator and the divisor as the denominator.
For example, we follow the offered measures to transform 7/3 into a blended number form.
Step 1: Divide 7 by 3
Tip 2: Write quotient, divisor and also remainder in form as in action 2 and step 3 over.
One have the right to include combined numbers by rearranging the entirety numbers, including them individually and adding the leftover fractions individually and also in the end combing them all.
11⁄2+ 33⁄4
Adding the whole numbers individually and the fractions separately.
For totality numbers:
1+3 = 4
For fractions: Find the LCM and then add
In the finish, including both the components together.
4+11⁄4=51⁄4
Real life examples
We deserve to inspect our knowledge of mixed fractions by expushing the parts of a totality as blended fractions while serving a pizza or a pie at house. Leftover pizzas, half-filled glasses of milk develop examples of mixed fractions.
Fun FactsMixed numbers are also referred to as combined fractions.See more: Why Is Meiosis Referred To As Reduction Division '? Attention Required!
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Weaning is the process of food consumption by the baby from the source other than breast milk. It is a way of complementing the breastfeeding with the other food sources as a nutrition supplement for the development of the baby. A majority of the doctors recommend exclusive breastfeeding until the baby is six months old. The period extending up to one year proves more beneficial for the nutrition fulfilment to the baby. World Health Organization recommends breastfeeding for two years after the birth of the baby. In a majority of the cases surveyed so far, weaning starts along with the breastfeeding.
The readiness among the babies for the weaning varies from one year to four years of age. An infant starts to eat a variety of foods. It begins with licking the food followed by gradual eating. The babies begin self-feeding at the age of 6 months which is a sign of their personal development. Their internal system matures at the same age making them ready for the weaning.
Here, we have few factors that determine the importance of the weaning.
- Oral motor skill development: An infant follows the pattern of suck-swallow-breathe since its birthday until the beginning of the weaning. In suck-swallow-breathe sequence, the oral motor functioning involves taking in the milk, swallowing it and breathing. With the weaning, the infant starts to learn other oral motor patterns that include tongue lateralization and elevation and chewing. Through oral motor development, the baby learns managing bolus formation and swallowing time. Thus, weaning helps the infant in developing oral motor skills.
- Maturation of digestive system: Many studies found that the maturation of the digestive system in an infant begins at the age of 4-6 months. This makes the internal system of an infant compatible with the solids. With the process of weaning, the infant’s digestive system develops and becomes compatible with digesting solid foods along with breast milk.
- Self-feeding: Weaning helps an infant with the initiation of self-feeding. An infant starts to explore food items, their textures, and tastes. Self-feeding allows an infant to make a personal choice in the food which proves as a beginning of its personal development. It encourages child’s motor development including eye-hand coordination and chewing.
- Beginning of crawling: The beginning of crawling is a boon offered by weaning. Weaning develops chewing ability and generates a likeliness toward a specific flavour among infants. Several research concluded that gaining of such an ability motivates an infant to crawl toward delicious food. Thus, weaning helps in the initiation of extended movement in an infant.
- Olfactory system improvement: Weaning activates an olfactory system in the baby. As weaning places the focus of the baby on the smell of the food along with colour, texture, and taste, the olfactory system starts to integrate with other senses and form a sense of flavour. This allows a child to develop smell identification and accept or reject a specific food item.
- Makes independent: Many reports show that weaning makes a child independent. At times, a child rejects the food offered to it but eats the same food by itself. Such instances prove that a child is willing to use its recent developed motor skills by choosing food according to its taste and self-feeding. Also, World Health Organization advice parents to encourage their baby to take advantage of its development stages.
- Social attachment: Feeding a baby becomes a family and social activity at every home. It keeps a happy and belonging environment at home. The baby starts to learn eating skills by watching and imitating family members and relatives and focuses on eating the same food as everyone else. In this way, weaning provides an opportunity to all the family members to have fun together while feeding the baby.
Weaning includes few guidelines that a parent must follow while considering the addition of solids to the baby’s diet. At the beginning of weaning, a parent must make her child sit upright in a highly supportive chair during all the meals. This will help the child in effective swallowing and choking reduction. Also, a correct sitting will support efficient oral motor coordination.
A general problem a parent face includes disinterest of a child in food. In such cases, a parent can develop an eating interest in a child by taking meals in front of the child. Such an action will motivate the child to consume the same food. Now, we can say that weaning is an important part of infant-hood, as it is the beginning of the overall development of the baby including its physical and mental development.<|endoftext|>
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# 14.2.6.1: Prelude to Sequence and Series
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The Koch snowflake1 is constructed from an infinite number of non-overlapping equilateral triangles. Consequently, we can express its area as a sum of infinitely many terms. How do we add an infinite number of terms? Can a sum of an infinite number of terms be finite? To answer these questions, we need to introduce the concept of an infinite series, a sum with infinitely many terms. Having defined the necessary tools, we will be able to calculate the area of the Koch snowflake.
The topic of infinite series may seem unrelated to differential and integral calculus. In fact, an infinite series whose terms involve powers of a variable is a powerful tool that we can use to express functions as “infinite polynomials.” We can use infinite series to evaluate complicated functions, approximate definite integrals, and create new functions. In addition, infinite series are used to solve differential equations that model physical behavior, from tiny electronic circuits to Earth-orbiting satellites.
1The Koch snowflake is a Koch curve which is a fractal curve. Fractals are seen in nature (snowflake for instance as well as trees, coastlines, broccoli, galaxies, etc.) but are also seen in constructions such as the Khajuraho temples in India (see below). Fractal analysis (many methods which are usually used together for robust comparison) which is an analysis that looks for fractal characteristics to extract information form a signal. One application of fractal analysis is cancer detection.
Note that there are different types of fractals. The Koch snowflake is fractal curve that lends itself to the discussion in this section, other fractals are not applicable to this section. Fractals are real mathematics/science but there are a numerous pseudo-science writings that use fractals in questionable ways. Follow the guidelines discussed herein before when reading about fractals.
A couple of the Khajuraho temples which have a fractal structure to them. Wikipedia image (Abinthomas0007, CC BY-SA 4.0,9/30/2017).<|endoftext|>
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In England today, you say “Happy Christmas!” In the United States, you say, “Merry Christmas!” Why the difference?
Believe it or not, the “Merry Christmas!” greeting we use in the US originated in England, more than four hundred years ago. It is first recorded in 1565 in the Hereford Municipal Manuscript, The author offered good wishes that God would send a “mery Christmas” to the readers.
English author Charles Dickens popularized the greeting in his 1843 story, “A Christmas Carol,” the focus of this year’s movie, The Man Who Invented Christmas.
The very first Christmas card printed for sale, also in 1843, used the words “Merry Christmas” as part of its greeting as well.
However, Queen Elizabeth II broke with her own nation’s tradition. She began wishing her subjects a “happy Christmas” in her annual holiday broadcasts more than 60 years ago.
People speculate that the Queen substituted the word “happy” because for many centuries the word “merry” had referred to people who were “happy” as a result of drinking too much alcohol. The Queen is reported to have moral scruples about overindulgence of any kind. Therefore, she does not encourage her subjects to make “merry” at Christmas or any other time.
Meanwhile, here in the US, we’ve completely forgotten that “merry” used to mean “intoxicated.” So, in the words of the song, “have yourself a merry little Christmas” – even if there’s no rum in the eggnog.
This post was first written on request of ESL teacher Cecelia Barker of Raleigh, NC, for use in her “Oral Production” class. The Grammar Queen was unfamiliar with “Oral Production” as an instructional topic and had to request an explanation.
The Grammar Queen learned that classes in “Oral Production” aid second language speakers in voicing a new language with the same nuanced sloppiness as natives — to elide words, for example, instead of pronouncing each separately and distinctly. Because the Grammar Queen speaks as a native, although writing as a professional, she withholds judgment about this casual verbal habit, to which she herself is not subject — the Queen is never subject — but is culturally habituated.<|endoftext|>
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What is DNA Sequencing?
DNA sequencing is essentially another term for “reading the DNA double helix” or determining the sequence of its nucleotides or bases. Nucleotides are comprised of four chemical bases: adenine (A), thymine (T), cytosine (C), and guanine (G). These bases always form the same base pairs within the DNA double helix: A always pairs with T, and C always pairs with G. This pairing is important for many processes in the cell, including the copying of DNA molecules during cell division. It is also fundamental in DNA sequencing.
About three billion base pairs of human DNA, the human genome, were sequenced as part of the Human Genome Project (HGP), a collaborative project involving an international team of researchers. The sequencing techniques that were developed to execute the project not only enabled the sequencing of most of the human genome, they also identified and mapped most of the genes within the human genome.
History and Different Types of DNA Sequencing
Allan Maxam and Walter Gilbert developed the first widely adopted method for DNA sequencing in 1973. In 1977, Frederick Sanger and his colleagues developed an alternative method, known as Sanger sequencing or the chain termination method.
Initially, the Maxam-Gilbert method was the more popular method, as purified DNA could be used directly—whereas the Sanger method required cloning to produce single-stranded DNA before sequencing. However, the Maxam-Gilbert method had a number of drawbacks including challenges in scaling up, the exposure of scientists using the method to hazardous chemicals, and technical complexity.
These drawbacks, in combination with the improvement of Sanger sequencing, ensured that Sanger's chain termination method became the most popular of the first generation sequencing methods. It remained widely used, with modifications, for decades.
In the mid-2000s a new kind of DNA sequencing technology known as next generation sequencing (NGS) emerged. It allowed for the sequencing of several DNA molecules in parallel, dramatically increasing the speed of DNA sequencing.
For example, 454 sequencing, one of the first next gen DNA sequencing technologies launched in 2005, made huge advances in terms of the rate of DNA sequencing. Researchers used this technology to sequence the genome of renowned scientist James Watson in just two months. By contrast, the Human Genome Project (HGP), completed in 2003, took 15 years.
The increases in speed and efficiency in sequencing have in turn resulted in much lower costs for the sequencing and analysis of large amounts of DNA. Whereas the cost of the HGP was $3 billion, scientists were able to sequence Watson’s genome for less than $1 million dollars. Today these costs have dropped to approximately $1,000 per genome.
A number of parallel sequencing techniques have been developed following 454 sequencing. While Sanger sequencing is still relatively widely used for smaller scale projects that focus on sequencing individual segments of DNA, today genomes are typically sequenced using these faster, less expensive parallel sequencing methods. And new sequencing methods are emerging all the time as DNA sequencing technologies continue to evolve.
What Does DNA Sequencing Tell Us?
Even though DNA sequencing technology is only about four decades old, its impact on the medical, scientific, and research fields is profound. The implications of DNA sequencing are vast and promising, with a great potential for everything from learning more about your history to diagnostic and therapeutic applications.
The challenge of NGS technologies will be analyzing the large amounts of data that will be available in the coming years as these DNA sequencing technologies become faster and more efficient, generating ever larger amounts of data.<|endoftext|>
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Even a single species of bacteria can positively affect soils and plants
Microbes deep in the soil influence plant health by releasing potent natural antibiotics such as PCA (phenazine-1-carboxylic acid). PCA-producing bacteria thrive on roots of dryland wheat throughout the Columbia Plateau, a major wheat-producing region in central Washington and Oregon, but their role in this important ecosystem was something of a mystery. Now the work of an international team of scientists provides direct evidence for the first time that these bacteria affect not only the wheat, but the soil around it.
Dry areas like the Columbia Plateau suffer large soil losses from wind erosion and plants often struggle to survive droughts. The availability of PCA encourages development of a biofilm that could combat soil degradation by improving water retention. This biofilm also protects the roots from drying out in drought conditions. Most importantly, PCA-producing bacteria enhance long-term soil health by contributing to soil organic matter. Understanding how these bacteria support the ecosystem may prove key to improving agriculture not only in the Columbia Plateau, but in dryland areas around the world.
Researchers set out to discover the mechanisms that control the accumulation of PCA under dryland conditions. Led by Melissa LeTourneau, an Office of Science Graduate Student Research (SCGSR) Fellow at Washington State University, the team included researchers from Pacific Northwest National Laboratory, University of Southern Mississippi, India's Institute of Bioresources and Sustainable Development, U.S. Department of Agriculture Agricultural Research Service, and EMSL, the Environmental Molecular Sciences Laboratory, a U.S. Department of Energy Office of Science user facility. The researchers compared the biofilms on roots inoculated with one strain of PCA-producing bacteria to biofilms on roots lacking PCA-producing bacteria. Examining the samples using a suite of highly advanced microscopes, including EMSL's new-generation ion microprobe (NanoSIMS), helium ion microscope, and focused ion beam/scanning electron microscope, they found PCA promotes biofilm development in dryland root systems and likely influences crop nutrition and soil health in dryland wheat fields. The results fill the gaps in our understanding of dynamics and effect of PCA in dryland agricultural ecosystems.<|endoftext|>
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# ML Aggarwal Solutions for Class 10 Maths Chapter 8 Matrices
ML Aggarwal Solutions For Class 10 Maths Chapter 8 Matrices consists of accurate solutions, which help the students to quickly complete their homework and prepare well for the exams. These Solutions of ML Aggarwal provide students an advantage with practical questions. Each step in the solution is explained to match students’ understanding. To score good marks in Class 10 Mathematics examination, it is advised they solve questions provided at the end of each chapter in the ML Aggarwal textbooks for Class 10. This chapter deals with matrices, types of matrices and product of matrices. A rectangular array of m x n numbers (real or complex) in the form of m horizontal lines (called rows) and n vertical lines (called columns), is called a matrix of order m by n, written as m x n matrix. Such an array is enclosed by [ ] or ( ). This chapter of ML Aggarwal Solutions for Class 10 contains three exercises with chapter test. These solutions provided by BYJU’S cover all these concepts, with detailed explanations.
## ML Aggarwal Solutions for Class 10 Maths Matrices:-
### Access answers to ML Aggarwal Solutions for Class 10 Maths Chapter 8 – Matrices
Exercise 8.1
1. Classify the following matrices:
Solution:
It is square matrix of order 2
Solution:
It is row matrix of order 1 × 3
Solution:
It is column matrix of order 3 × 1
Solution:
It is a matrix of order 3 × 2
Solution:
It is a matrix of order 2 × 3
Solution:
It is zero matrix of order 2 × 3
2. (i) If a matrix has 4 elements, what are the possible order it can have?
Solution:
It can have 1 × 4, 4 × 1 or 2 × 2 order.
(ii) If a matrix has 4 elements, what are the possible orders it can have?
Solution:
It can have 1 × 8, 8 × 1, 2 × 4 or 4 × 2 order.
3. Construct a 2 × 2 matrix whose elements aij are given by
(i) aij = 2i – j
(ii) aij =i.j
Solution:
(i) Given aij = 2i – j
Therefore matrix of order 2 × 2 is
(ii) Given aij =i.j
Therefore matrix of order 2 × 2 is
4. Find the values of x and y if:
Solution:
Given
Now by comparing the corresponding elements,
2x + y = 5 ….. i
3x – 2y = 4 ….ii
Multiply (i) by 2 and (ii) by 1 we get
4x + 2y = 10 and 3x – 2y = 4
7x = 14
x = 14/7
x = 2
Substituting the value of x in (i)
4 + y = 5
y = 5 – 4
y = 1
Hence x = 2 and y = 1
5. Find the value of x if
Solution:
Given
Comparing the corresponding terms of given matrix we get
-y = 2
Therefore y = -2
Again we have
3x + y = 1
3x = 1 – y
Substituting the value of y we get
3x = 1 – (-2)
3x = 1 + 2
3x = 3
x = 3/3
x = 1
Hence x = 1 and y = -2
6. If
Find the values of x and y.
Solution:
Given
Comparing the corresponding terms, we get
x + 3 = 5
x = 5 – 3
x = 2
Again we have
y – 4 = 3
y = 3 + 4
y = 7
Hence x = 2 and y = 7
7. Find the values of x, y and z if
Solution:
Given
Comparing the corresponding elements of given matrix, then we get
x + 2 = -5
x = -5 – 2
x = -7
Also we have 5z = -20
z = -20/5
z = – 4
Again from given matrix we have
y2 + y – 6 = 0
The above equation can be written as
y2 + 3y – 2y – 6 = 0
y (y + 3) – 2 (y + 3) = 0
y + 3 = 0 or y – 2 = 0
y = -3 or y = 2
Hence x = -7, y = -3, 2 and z = -4
8. Find the values of x, y, a and b if
Solution:
Given
Comparing the corresponding elements
x – 2 = 3 and y = 1
x = 2 + 3
x = 5
again we have
a + 2b = 5….. i
3a – b = 1 ……ii
Multiply (i) by 1 and (ii) by 2
a + 2b = 5
6a – 2b = 2
Now by adding above equations we get
7a = 7
a = 7/7
a = 1
Substituting the value of a in (i) we get
1 + 2b = 5
2b = 5 -1
2b = 4
b = 4/2
b = 2
9. Find the values of a, b, c and d if
Solution:
Given
Comparing the corresponding terms, we get
3 = d
d = 3
Also we have
5 + c = -1
c = -1 – 5
c = -6
Also we have,
a + b = 6 and a b = 8
we know that,
(a – b)2 = (a + b)2 – 4 ab
(6)2 – 32 = 36 – 32 = 4 = (± 2)2
a – b = ± 2
If a – b = 2
a + b = 6
Adding the above two equations we get
2a = 4
a = 4/2
a = 2
b = 6 – 4
b = 2
Again we have a – b = -2
And a + b = 6
2a = 4
a = 4/2
a = 2
Also, b = 6 – 2 = 4
a = 2 and b = 4
Exercise 8.2
Solution:
Solution:
3. Simplify:
Solution:
Given,
4.
Solution:
5.
Find the matrix X if:
(i) 3A + X = B
(ii) X – 3B = 2A
Solution:
6. Solve the matrix equation
Solution:
7.
Solution:
8.
Solution:
9.
Solution:
10.
Solution:
On comparing the corresponding elements, we have
8 + y = 0
Then, y = -8
And, 2x + 1 = 5
2x = 5 – 1 = 4
x = 4/2 = 2
Therefore, x = 2 and y = -8
11.
Solution:
On comparing the corresponding terms, we have
2x + 1 = 5
2x = 5 -1 = 4
x = 4/2 = 2
And,
8 + y = 0
y = -8
And, z = 7
Therefore, x = 2, y = -8 and z = 7.
12.
Solution:
Now, comparing the corresponding terms, we get
4 – 4x = -8
4 + 8 = 4x
12 = 4x
x = 12/4
x = 3
And, y + 5 = 2
y = 2 – 5 =
y = -3
Therefore, x = 3 and y = -3
13.
Find the value of a, b and c.
Solution:
Next, on comparing the corresponding terms, we have
a + 1 = 5 ⇒ a = 4
b + 2 = 0 ⇒ b = -2
-c = 3 ⇒ c = -3
Therefore, the value of a, b and c are 4, -2 and -3 respectively.
14. and 5A + 2B = C, find the values of a, b and c.
Solution:
On comparing the corresponding terms, we get
5a + 6 = 9
5a = 9 – 6
5a = 3
a = 3/5
And,
25 + 2b = -11
2b = -11 – 25
2b = -36
b = -36/2
b = -18
And, c = 6
Therefore, the value of a, b and c are 3/5, -18 and 6 respectively.
Exercise 8.3
Solution:
Yes, the product is possible because of number of column in A = number of row in B
That is order of matrix is 2 × 1
Solution:
3.
Solution:
4.
Solution:
5. Given matrices:
Find the products of
(i) ABC
(ii) ACB and state whether they are equal.
Solution:
Now consider,
6.
Solution:
Given
7.
Solution:
Solution:
(i) A(B + C) (ii) (B + C)A
Solution:
10.
Find the matrix C(B – A).
Solution:
Given,
Find A2 + AB + B2.
Solution:
Given,
12. , find A2 + AC – 5B.
Solution:
Given,
13. If , find AC + B2 – 10C.
Solution:
Given,
14. If find A2 and A3. Also state that which of these is equal to A.
Solution:
Given,
From above, its clearly seen that A3 = A.
15. If show that 6X – X2 = 9I where I is the unit matrix.
Solution:
Given,
– Hence proved
16. Show that is a solution of the matrix equation X2 – 2X – 3I = 0, where I is
the unit matrix of order 2.
Solution:
Given,
17. Find the matrix 2 × 2 which satisfies the equation
Solution:
Given,
18. If find the value of x, so that A2 – 0
Solution:
Given,
On comparing,
1 + x = 0
∴ x = -1
19.
Solution:
Comparing the corresponding elements,
– 3x + 4 = -5
-3x = -5 – 4 = -9
x = -9/-3 = 3
Therefore, x = 3 and y = -10.
Comparing, we get
8x = 16
⇒ x = 16/8 = 2
And, 9y = 9
y = 9/9 = 1
20. Find the values of x and y if
Solution:
Given,
On comparing the corresponding elements, we have
2x + y = 3 … (i)
3x + y = 2 … (ii)
Subtracting, we get
-x = 1 ⇒ x = -1
Substituting the value of x in (i),
2(-1) + y = 3
-2 + y = 3
y = 3 + 2 = 5
Therefore, x = -1 and y = 5.
Chapter test
Solution:
Given
comparing the corresponding elementsÂ
a + 3 = 2a + 1Â
⇒ 2a – a =3 – 1Â
⇒ a = 2 b² + 2 = 3bÂ
⇒ b² – 3b + 2 = 0Â
⇒ b² – b – 2b + 2 = 0Â
⇒ b (b – 1) – 2 (b – 1) = 0Â
⇒ (b – 1) (b – 2) = 0.Â
Either b – 1 = 0,
then b = 1 or b – 2 = 0,
then b = 2Â
Hence a = 2, b = 2 or 1
Solution:
Now comparing the corresponding elements
3a = 4 + a
a – a = 4
2a = 4
Therefore, a = 2
3b = a + b + 6
3b – b = 2 + 6
2b = 8
Therefore, b = 4
3d = 3 + 2d
3d – 2d = 3
Therefore, d = 3
3c = c + d – 1
3c – c = 3 – 1
2c = 2
Therefore, c = 1
Hence a = 2,b = 4, c = 1 and d= 3
3. Determine the matrices A and B when
Solution:
Given,
4.
Solution:
Comparing the corresponding elements, we have
4 + 2a = 18
2a = 18 – 4 = 14
a = 14/2
⇒ a = 7
1 + 2b = 7
2b = 7 – 1 = 6
b = 6/2
⇒ b = 3
2 + 2c = 14
2c = 14 – 2 = 12
2c = 12
c = 12/2
⇒ c = 6
3 + 2d = 11
2d = 11 – 3
d = 8/2
⇒ d = 4
Therefore, a = 7, b = 3, c = 6 and d = 4.
5. If , find the each of the following and state it they
are equal:
(i) (A + B) (A – B)
(ii) A2 – B2
Solution:
Given,
Hence, its clearly seen that (A + B) (A – B) ≠A2 – B2.
6. If , find A2 – 5A – 14I, where I is unit matrix of order 2 × 2.
Solution:
Given,
7. If and A2 = 0, find p and q.
Solution:
On comparing the corresponding elements, we have
9 + 3p = 0
3p = -9
p = -9/3
p = -3
And,
9 + 3q = 0
3q = -9
q = -9/3
q = -3
Therefore, p = -3 and q = -3.
8. If find a, b, c and d.
Solution:
Given,
On comparing the corresponding elements, we have
-a = 1 ⇒ a = -1
-b = 0 ⇒ b = 0
c = 0 and d = -1
Therefore, a = -1, b = 0, c = 0 and = -1.
9. Find a and b if
Solution:
On comparing the corresponding terms, we have
2a – 4 = 0
2a = 4
a = 4/2
a = 2
And, 2a – 2b = -2
2(2) – 2b = -2
4 – 2b = -2
2b = 4 + 2
b = 6/2
b = 3
Therefore, a = 2 and b = 3.
10. If
Find (i) 2A – 3B (ii) A2 (iii) BA
Solution:<|endoftext|>
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PreTeXt Sample Book: Abstract Algebra (SAMPLE ONLY)
AppendixBHints and Answers to Selected Odd Exercises
2The Integers2.4Exercises
2.4.1.
The base case, $$S(1): [1(1 + 1)(2(1) + 1)]/6 = 1 = 1^2$$ is true.
Assume that $$S(k): 1^2 + 2^2 + \cdots + k^2 = [k(k + 1)(2k + 1)]/6$$ is true. Then
\begin{align*} 1^2 + 2^2 + \cdots + k^2 + (k + 1)^2 & = [k(k + 1)(2k + 1)]/6 + (k + 1)^2\\ & = [(k + 1)((k + 1) + 1)(2(k + 1) + 1)]/6, \end{align*}
and so $$S(k + 1)$$ is true. Thus, $$S(n)$$ is true for all positive integers $$n\text{.}$$
2.4.3.
The base case, $$S(4): 4! = 24 \gt 16 =2^4$$ is true. Assume $$S(k): k! \gt 2^k$$ is true. Then $$(k + 1)! = k! (k + 1) \gt 2^k \cdot 2 = 2^{k + 1}\text{,}$$ so $$S(k + 1)$$ is true. Thus, $$S(n)$$ is true for all positive integers $$n\text{.}$$
2.4.11.
Hint.
The base case, $$S(0): (1 + x)^0 - 1 = 0 \geq 0 = 0 \cdot x$$ is true. Assume $$S(k): (1 + x)^k -1 \geq kx$$ is true. Then
\begin{align*} (1 + x)^{k + 1} - 1 & = (1 + x)(1 + x)^k -1\\ & = (1 + x)^k + x(1 + x)^k - 1\\ & \geq kx + x(1 + x)^k\\ & \geq kx + x\\ & = (k + 1)x, \end{align*}
so $$S(k + 1)$$ is true. Therefore, $$S(n)$$ is true for all positive integers $$n\text{.}$$
2.4.19.
Hint.
Use the Fundamental Theorem of Arithmetic.
2.4.23.
Hint.
Let $$S = \{s \in {\mathbb N} : a \mid s\text{,}$$ $$b \mid s \}\text{.}$$ Then $$S \neq \emptyset\text{,}$$ since $$|ab| \in S\text{.}$$ By the Principle of Well-Ordering, $$S$$ contains a least element $$m\text{.}$$ To show uniqueness, suppose that $$a \mid n$$ and $$b \mid n$$ for some $$n \in {\mathbb N}\text{.}$$ By the division algorithm, there exist unique integers $$q$$ and $$r$$ such that $$n = mq + r\text{,}$$ where $$0 \leq r \lt m\text{.}$$ Since $$a$$ and $$b$$ divide both $$m\text{,}$$ and $$n\text{,}$$ it must be the case that $$a$$ and $$b$$ both divide $$r\text{.}$$ Thus, $$r = 0$$ by the minimality of $$m\text{.}$$ Therefore, $$m \mid n\text{.}$$
2.4.27.
Hint.
Since $$\gcd(a,b) = 1\text{,}$$ there exist integers $$r$$ and $$s$$ such that $$ar + bs = 1\text{.}$$ Thus, $$acr + bcs = c\text{.}$$ Since $$a$$ divides both $$bc$$ and itself, $$a$$ must divide $$c\text{.}$$
2.4.29.
Hint.
Every prime must be of the form 2, 3, $$6n + 1\text{,}$$ or $$6n + 5\text{.}$$ Suppose there are only finitely many primes of the form $$6k + 5\text{.}$$
3Groups3.5Exercises
3.5.1.
Hint.
(a) $$3 + 7 \mathbb Z = \{ \ldots, -4, 3, 10, \ldots \}\text{;}$$ (c) $$18 + 26 \mathbb Z\text{;}$$ (e) $$5 + 6 \mathbb Z\text{.}$$
3.5.15.
Hint.
There is a nonabelian group containing six elements.
3.5.17.
Hint.
The are five different groups of order 8.
3.5.25.
Hint.
\begin{align*} (aba^{-1})^n & = (aba^{-1})(aba^{-1}) \cdots (aba^{-1})\\ & = ab(aa^{-1})b(aa^{-1})b \cdots b(aa^{-1})ba^{-1}\\ & = ab^na^{-1}. \end{align*}
3.5.31.
Hint.
Since $$abab = (ab)^2 = e = a^2 b^2 = aabb\text{,}$$ we know that $$ba = ab\text{.}$$
3.5.35.
Hint.
$$H_1 = \{ id \}\text{,}$$ $$H_2 = \{ id, \rho_1, \rho_2 \}\text{,}$$ $$H_3 = \{ id, \mu_1 \}\text{,}$$ $$H_4 = \{ id, \mu_2 \}\text{,}$$ $$H_5 = \{ id, \mu_3 \}\text{,}$$ $$S_3\text{.}$$
3.5.41.
Hint.
The identity of $$G$$ is $$1 = 1 + 0 \sqrt{2}\text{.}$$ Since $$(a + b \sqrt{2}\, )(c + d \sqrt{2}\, ) = (ac + 2bd) + (ad + bc)\sqrt{2}\text{,}$$ $$G$$ is closed under multiplication. Finally, $$(a + b \sqrt{2}\, )^{-1} = a/(a^2 - 2b^2) - b\sqrt{2}/(a^2 - 2 b^2)\text{.}$$
3.5.49.
Hint.
Since $$a^4b = ba\text{,}$$ it must be the case that $$b = a^6 b = a^2 b a\text{,}$$ and we can conclude that $$ab = a^3 b a = ba\text{.}$$
3.5.55.
$$1$$
3.5.57.
$$n$$
3.5.59.
3.5.59.a
$$2$$
3.5.59.b
3.5.59.b.i
$$6$$
3.5.59.b.ii
$$10$$
5Runestone Testing5.7True/False Exercises
5.7.1.True/False.
Hint.
$$P_n\text{,}$$ the vector space of polynomials with degree at most $$n\text{,}$$ has dimension $$n+1$$ by Theorem 3.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter $$n\text{?}$$
5.8Multiple Choice Exercises
Hint 1.
What did you see last time you went driving?
Hint 2.
Maybe go out for a drive?
Hint 1.
What did you see last time you went driving?
Hint 2.
Maybe go out for a drive?
5.8.5.Mathematical Multiple-Choice, Not Randomized, Multiple Answers.
Hint.
You can take a derivative on any one of the choices to see if it is correct or not, rather than using techniques of integration to find a single correct answer.
5.9Parsons Exercises
5.9.1.Parsons Problem, Mathematical Proof.
Hint.
Dorothy will not be much help with this proof.
5.11Matching Exercises
5.11.3.Matching Problem, Linear Algebra.
Hint.
For openers, a basis for a subspace must be a subset of the subspace.
5.12Clickable Area Exercises
5.12.3.Clickable Areas, Text in a Table.
Hint.
Python boolean variables begin with capital latters.
5.16Hodgepodge>
5.16.1.With Tasks in an Exercises Division.
5.16.1.aTrue/False.
Hint.
$$P_n\text{,}$$ the vector space of polynomials with degree at most $$n\text{,}$$ has dimension $$n+1$$ by Theorem 3.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter $$n\text{?}$$
5.17Exercises that are Timed
Timed Exercises
5.17.1.True/False.
Hint.
$$P_n\text{,}$$ the vector space of polynomials with degree at most $$n\text{,}$$ has dimension $$n+1$$ by Theorem 3.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter $$n\text{?}$$<|endoftext|>
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# Fraction calculator
The calculator performs basic and advanced operations with fractions, expressions with fractions combined with integers, decimals, and mixed numbers. It also shows detailed step-by-step information about the fraction calculation procedure. Solve problems with two, three, or more fractions and numbers in one expression.
## Result:
### (2/5) : (1/6) = 12/5 = 2 2/5 = 2.4
Spelled result in words is twelve fifths (or two and two fifths).
### How do you solve fractions step by step?
1. Divide: 2/5 : 1/6 = 2/5 · 6/1 = 2 · 6/5 · 1 = 12/5
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 1/6 is 6/1) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the next intermediate step, the fraction result cannot be further simplified by canceling.
In words - two fifths divided by one sixth = twelve fifths.
#### Rules for expressions with fractions:
Fractions - use the slash “/” between the numerator and denominator, i.e., for five-hundredths, enter 5/100. If you are using mixed numbers, be sure to leave a single space between the whole and fraction part.
The slash separates the numerator (number above a fraction line) and denominator (number below).
Mixed numerals (mixed fractions or mixed numbers) write as non-zero integer separated by one space and fraction i.e., 1 2/3 (having the same sign). An example of a negative mixed fraction: -5 1/2.
Because slash is both signs for fraction line and division, we recommended use colon (:) as the operator of division fractions i.e., 1/2 : 3.
Decimals (decimal numbers) enter with a decimal point . and they are automatically converted to fractions - i.e. 1.45.
The colon : and slash / is the symbol of division. Can be used to divide mixed numbers 1 2/3 : 4 3/8 or can be used for write complex fractions i.e. 1/2 : 1/3.
An asterisk * or × is the symbol for multiplication.
Plus + is addition, minus sign - is subtraction and ()[] is mathematical parentheses.
The exponentiation/power symbol is ^ - for example: (7/8-4/5)^2 = (7/8-4/5)2
#### Examples:
adding fractions: 2/4 + 3/4
subtracting fractions: 2/3 - 1/2
multiplying fractions: 7/8 * 3/9
dividing Fractions: 1/2 : 3/4
exponentiation of fraction: 3/5^3
fractional exponents: 16 ^ 1/2
adding fractions and mixed numbers: 8/5 + 6 2/7
dividing integer and fraction: 5 ÷ 1/2
complex fractions: 5/8 : 2 2/3
decimal to fraction: 0.625
Fraction to Decimal: 1/4
Fraction to Percent: 1/8 %
comparing fractions: 1/4 2/3
multiplying a fraction by a whole number: 6 * 3/4
square root of a fraction: sqrt(1/16)
reducing or simplifying the fraction (simplification) - dividing the numerator and denominator of a fraction by the same non-zero number - equivalent fraction: 4/22
expression with brackets: 1/3 * (1/2 - 3 3/8)
compound fraction: 3/4 of 5/7
fractions multiple: 2/3 of 3/5
divide to find the quotient: 3/5 ÷ 2/3
The calculator follows well-known rules for order of operations. The most common mnemonics for remembering this order of operations are:
PEMDAS - Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
BEDMAS - Brackets, Exponents, Division, Multiplication, Addition, Subtraction
BODMAS - Brackets, Of or Order, Division, Multiplication, Addition, Subtraction.
GEMDAS - Grouping Symbols - brackets (){}, Exponents, Multiplication, Division, Addition, Subtraction.
Be careful, always do multiplication and division before addition and subtraction. Some operators (+ and -) and (* and /) has the same priority and then must evaluate from left to right.
## Fractions in word problems:
• Unit rate
Find unit rate: 6,840 customers in 45 days
• The third
The one-third rod is blue, one-half of the rod is red, the rest of the rod is white and measures 8 cm. How long is the whole rod?
• Sewing
Beth's mother can sew 235 pairs of short pants in 6 days while Lourdes can sew 187 pairs in 8 days. How many more pairs of short pants can Beth's mother sew?
• Pumps
6 pump fills the tank for 3 and a half days. How long will fill the tank 7 equally powerful pumps?
• Almonds
Rudi has 4 cups of almonds. His trail mix recipe calls for 2/3 cup of almonds. How many batches of trail mix can he make?
• How many 16
How many three-tenths are there in two and one-fourths?
• Golf balls
Of the 28 golf balls, 1/7 are yellow. How many golf balls are yellow? Use the model to help you. Enter your answer in the box.
• Cutting wire
If you cut a 3 ½ ft length wire into pieces that are 2 inches long, how many pieces of wire will you have?
• The balls
You have 108 red and 180 green balls. You have to be grouped into the bags so that the ratio of red to green in each bag was the same. What smallest number of balls may be in one bag?
• Statues
Diana is painting statues. She has 7/8 of a liter of paint remaining. Each statue requires 1/20 of a liter of paint. How many statues can she paint?
• Video game
Nicole is playing a video game where each round lasts 7/12 of an hour. She has scheduled 3 3/4 hours to play the game. How many rounds can Nicole play?
• Jordan
Jordan wants to bring in cookie cakes to share with his 23 classmates for his birthday. His mom is willing to buy no more than 6 cookie cakes and Jordan must share the cakes equally between his classmates (and himself). Answer the questions to help Jordan
• Christmas
Calculate how much of the school year (202 days long) take Christmas holidays 19 days long. Expressed as a decimal number and as a percentage.<|endoftext|>
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# Difference between revisions of "The Fourth Dimension"
## Explorations
Begin learning about flatland and the fourth dimesion with:
## Introduction to Dimensions
Malevich. The Knife grinder.
Webster’s Dictionary gives a description of dimensions 1, 2 , 3, and 4. We will also give a description of the 0th dimension.
• Space of zero dimensions: A space that has no length breadth or thickness. An example is a point.
• Space of one dimension: A space that has length but no breadth or thickness; a straight or curved line.
• Space of two dimensions: A space which has length and breadth, but no thickness; a plane or curved surface.
• Space of three dimensions: A space which has length, breadth, and thickness; a solid.
• Space of four dimensions: A kind of extension, which is assumed to have length, breadth, thickness, and also a fourth dimension.
Spaces of five, six, or more dimensions can also be studied using mathematics.
You have seen examples of 0-, 1-, 2-, and 3-dimensional objects all your life. Some examples:
• A pin-prick can be thought of as 0-dimensional. To the naked eye it has no length, width or height.
• When you draw a line on a piece of paper, you are drawing a representation of a 1-dimensional object. We only measure one direction. We will ask for the length of a line-segment, but we would not ask questions about its width or height. We think of a line as simply not having any width or height.
• When you measure the area of a geometric object you are measuring something 2-dimensional. Think about how you would find the area of a rectangle. You would measure the length and the width and multiply the two, right? You measure the two dimensions, and so we think of the rectangle as 2-dimensional.
• Anything 3-dimensional will require 3 measurements. Hence the volume of a box is considered 3-dimensional. It has length, width, and height.
• Consider making an appointment with someone at a 10 story building. You will have to tell them where to meet and when. The location is 3-dimensional, because we need 3 coordinates to find a place in space: longitude, latitude and height. But this means that to determine our place and time of meeting we require 4 pieces of information: longitude, latitude, height AND time. This means that our appointment is something 4-dimensional.
This idea of space and time, appropriately named 4-dimensional space-time, was used by Einstein when he brought forth his theory of relativity.
A point (0-dim), a segment (1-dim), a square (2 dim) and a cube (3-dim)
In the image above we see a point, a line segment, a square and a cube. These shapes are 0-, 1-, 2-, and 3-dimansional respectively. One could ask: what is the next shape in that sequence? In other words, is there something like a 4-cube.
To construct a model of this 4-cube, also known as a tesseract, we can follow the procedure of creating higher dimensional shapes. Think about extending the shape we have into a dimension perpendicular to those we already have. For instance, starting with a line segment we can draw an exact copy of the segment in the plane and then connect the corresponding vertices. This creates a square.
Using two copies of a segment (1-dim) we construct a square (2-dim) by connecting the corresponding vertices
Similarly, connecting corresponding vertices on two copies of the square will result in a cube.
Given two cubes we can connect the corresponding vertices and construct a 4-dimensional cube or tesseract.
A 4-dimensional cube. The figure on the right shows the two cubes (yellow and blue) whose corresponding vertices have been connected.
## Escher and Dimension
Escher looked at the interplay between 3-dimensional objects and their 2-dimensional depictions. He used the play on dimensions to create several interesting prints. Some examples include:
A famous print by Escher showing the contrast between 2 and 3 dimensions is the print named drawing hands. The hands in the print are clearly 3-dimensional. The hands and the pencils are shown as existing in space. It gets more interesting when we move our eye to the wrists and the lower arm. Here Escher transitions to a 2-dimensional image. The underlying piece of paper is depicted as entirely flat.
Escher's tessellations are all 2-dimensional. He referred to them as "Regular Divisions of the Plane" (Regelmatige Vlakverdeelingen in Dutch) and they all depict patterns that decorate a nice flat, 2-dimensional surface.
Escher also studied regular 3-dimensional shapes. Some examples include:
First, there are the platonic solids. If we experiment with regular polygons and try to build 3-dimensional shapes, then there are only 3 regular polygons that can be used by themselves.
The triangle can be used in three different ways, while the square and the pentagon result in two more platonic solids:
• Four triangles will form a tetrahedron.
• Eight triangles will form a octahedron.
• Twenty triangles will form an icosahedron.
• Six squares will form a cube.
• Twelve regular pentagons will form a dodecahedron.
In "Stars" we see two chameleons inside a shape made up of three octahedra. In the background - floating around in space - are some platonic solids and several other geometric 3-dimensional figures. Wikipedia has a short article with links describing some of the more exotic geometric shapes that are shown in the background. [Stars (M. C. Escher)]
## Flatland
Flatland was written in the 19th century, and is both a satire on Victorian Society and an exploration of the mathematical notion of dimensions. We read this story to develop some ideas about how to think about the 4th dimension. Even though we do live in the 4-dimensional space-time, most people are not comfortable with the 4th dimension at first. There are two questions we are interested in. What would a 4-dimensional being look like if it interacted with us? What would our 3-dimensional world look like if someone moved us into the 4th dimension? One way to think through these questions is to first ask them with all the dimensions dropped down a bit. What would a 3-dimensional being look like if it interacted with 2-dimensional beings (i.e. Flatlanders)? Or, what would a 2-dimensional being look like if it interacted with a 1-dimensional being? What would the 2-dimensional world look like if someone moved a Flatlander into the 3rd dimension? Abbott answers all of these questions in the book Flatland.
The 3-dimensional sphere appeared one 2-dimensional slice at a time. The sphere would first appear as a dot, and then grow into ever increasingly large circles. After reaching its biggest circumference, the circles would shrink back down to a point again. But the important part here is that the flatlanders could only see a 2-dimensional cross-section. Their 2-dimensional eyes and brains were not used to thinking about or seeing 3-dimensional beings.
Similarly, we would expect to see 3-dimensional cross-sections of any 4-dimensional beings.
When A. Square (the main character in Flatland) traveled to Lineland, he could see all of their world at once. The King of Lineland at first doesn’t know who is talking to him, because he can’t see Mr. A. Square at all. Later in the story A. Square moves into Spaceland, and is able to look down upon his own world.
A. Square is able to see the interior and exterior of his house at the same time. He can also see his family moving about the house. If you look carefully at the picture, you would see that A. Square can also see inside his relatives. Similarly, if we were moved into the 4th dimension, we might be able to see our world all at once. We would see the interior and exterior of our houses simultaneously, and we would also be able to see all around people.
## The fourth dimension in art
Several well known artists have explored the concept of the fourth dimension, while the work of some others can certainly be interpreted as depicting the fourth dimension.
Crucifixion (Corpus Hypercubus) - 1954 A Propos of the Treatise on Cubic Form by Juan de Herrera - 1960
Salvador Dali explored the 4-dimensional cube in two of his works. In 1954 he painted Crucifixion (Corpus Hypercubus) depicting Jesus crucified on the net of a hypercube. Dali's wife Gala is shown as the woman appearing before the cross.
In the painting A Propos of the Treatise on Cubic Form by Juan de Herrera from 1960, Dali gives a nod to the 16th century Spanish architect, mathematician and geometer Juan de Herrera. The painting shows a tesseract: two cubes with the corresponding vertices connected. The cube on the outside shows the foundation stone of the St. John Apostle and Evangelist Church in Northern Spain on four of the sides. The cube on the interior is a normal cube, while the edges connecting the corresponding vertices spell out the name of Juan de Herrera.
Other artists have depicted movement in a way that suggests space-time: a 3-dimensional object moves over time.
Nude descending a staircase by Marcel Duchamp (1912) Dynamism of a Dog on a Leash by Giacomo Balla (1912) The Knife Grinder by Kazimir Malevich (1912)<|endoftext|>
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RULES FOR COLONS
1. Use a colon when making a list, when what precedes the list is an independent clause.
CORRECT: There are four ingredients necessary for a good party: music, lighting, food, and personality.
There are four ingredients can stand alone, so the colon separates it from the list.
Do not use a colon to separate a preposition from its objects or a verb from its complements, since the clause will not be independent.
INCORRECT: My ancestors came from: Poland, Russia, and Ukraine.
The colon incorrectly separates the preposition from its objects, and leaves My ancestors came from, which is not an independent clause. Instead, the sentence should read:
CORRECT: My ancestors came from Poland, Russia, and Ukraine.
INCORRECT: The boys ran home and ate: cake, cookies, and soda.
The colon incorrectly separates the verb ate from its complements. The sentence should read:
CORRECT: The boys ran home and ate cake, cookies, and soda.
2. Use a colon after a complete statement to introduce related ideas:
CORRECT: The coffee shop is the best on the block: it has great scones, a full menu, and a great atmosphere.
If, when you are writing your essay, you are in doubt about whether or not you may use a colon, it is best to play it safe and separate your statements with a semi-colon or a period.<|endoftext|>
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# Matrix multiplication
#### Everything You Need in One Place
Homework problems? Exam preparation? Trying to grasp a concept or just brushing up the basics? Our extensive help & practice library have got you covered.
#### Learn and Practice With Ease
Our proven video lessons ease you through problems quickly, and you get tonnes of friendly practice on questions that trip students up on tests and finals.
#### Instant and Unlimited Help
Our personalized learning platform enables you to instantly find the exact walkthrough to your specific type of question. Activate unlimited help now!
##### Intros
###### Lessons
1. Multiplying a matrix by a matrix overview:
Dot product
##### Examples
###### Lessons
1. Dot product
Find the dot product of the following ordered $n$-tuples:
1. $\vec{a}=(2,4,6)$ and $\vec{b}=(1,3,5)$
2. $\vec{a}=(1,7,5,3)$ and $\vec{b}=(-2,3,6,1)$
3. $\vec{a}=(1,2)$ and $\vec{b}=(3,5)$
4. $\vec{a}=(7,-2,-1,4)$ and $\vec{b}=(1,1,2,2)$
2. Multiplying matrices
Multiply the following matrices:
3. Multiplying matrices with different dimensions
Multiply the following matrices:
###### Topic Notes
In this lesson, we will learn how to multiply a matrix with another matrix. But we will learn about n-tuples first. An n-tuple is an ordered list of n numbers. Multiplying an n-tuple by another n-tuple is called the dot product. The dot product is the summation of all product of each corresponding entries. To multiply a matrix with another matrix, we have to think of each row and column as a n-tuple. Each entry will be the dot product of the corresponding row of the first matrix and corresponding column of the second matrix. For example, if your entry is at the 3rd row and 4th column, then you have to take the dot product of the 3rd row of the first matrix and 4th column of the second matrix. Note that not all matrices can be multiplied.
## Matrix Multiplication
There are exactly two ways of multiplying matrices. The first way is to multiply a matrix with a scalar. This is known as scalar multiplication. The second way is to multiply a matrix with another matrix. That is known as matrix multiplication.
## Scalar Multiplication
scalar multiplication is actually a very simple matrix operation. To multiply a scalar with a matrix, we simply take the scalar and multiply it to each entry in the matrix. Let's do an example.
Question 1: Calculate $2A$ if
The question is asking us to find out what $2A$ is. In other words, we are finding
Notice that if we are to multiply 2 to each entry in the matrix, we get that:
Very simple! Let's do another one.
Question 2: Calculate $0A$ if
Again, we are trying to find $0A$. This means that we will be looking for the answer to
The matrix will be oddly shaped, but the concept remains the same. We will still multiple the scalar 0 to each entry in the matrix. Doing so gives us:
Notice that all the entries in the matrix are 0. This is known as a zero matrix that is 3 x 2.
Now that we are very familiar with scalar multiplication, why don't we move on to matrix multiplication?
## How to Multiply Matrices
To multiply a matrix by another matrix, we first need to learn about what dot product is.
## What is dot product?
Dot product (also known as vector multiplication) is a way to calculate the product of two vectors. For example, let the two vectors be:
How would I multiply these two vectors? Simply just multiply the corresponding entries, and add the products together. In other words,
So we get a single value from multiplying vectors. However, notice how that the two vectors have the same number of entries.What if one of the vectors has a different number of entries than the other? For example, let
If I were to multiply the corresponding entries and add them up all together, then I get:
There is a problem here. The first three entries have corresponding entries to multiply with, but the last entry doesn't. So what do we do here? The answer is we cannot do anything here. This just means we cannot calculate the dot product of these two vectors.
So in conclusion, we cannot find the dot product of two vectors that have different numbers of entries. They must have the same number of entries.
## 2 x 2 Matrix Multiplication
So what was the point of learning the dot product? Well, we will be using the dot product when we multiply two matrices together. When multiplying a matrix with another matrix, we want to treat rows and columns as a vector. More specifically, we want to treat each row in the first matrix as vectors, and each column in the second matrix as vectors. Let's do an example.
Question 3: Find $A \bullet B$ if
Multiplying the two matrices will give us:
Now the rows and the columns we are focusing are
where $r_{1}$ is the first row, $r_{2}$ is the second row, and, $c_{1}, c_{2}$ are first and second columns. Now we are going to treat each row and column we see here as a vector.
Notice here that multiplying a 2 x 2 matrix with another 2 x 2 matrix gives a 2 x 2 matrix. In other words, the matrix we get should have 4 entries.
How do we exactly get the first entry? Well, notice that the first entry is located on the first row and first column. So we simply take the dot product of $r_{1}$ and $c_{1}$. Thus, the first entry will be
How do we get the second entry this time? Well, notice that the location of the second entry is in the first row and second column. So we simply take the dot product of $r_{1}$ and $c_{2}$. Thus, the second entry will be
Now we are going to use the same strategy to look for the last two entries. Notice the second last entry is located in the $2^{nd}$ row and $1^{st}$ column, and the last entry is located in the $2^{nd}$ row and $2^{nd}$ column. So we take the dot product of $r_{2}$ and $c_{1}$, and the dot product of $r_{2}$ and $c_{2}$. This gives us:
Now we are done! This is what we get when we are multiplying 2 x 2 matrices. In general, the matrix multiplication formula for 2 x 2 matrices is
## 3x3 Matrix Multiplication
Now the process of a 3 x 3 matrix multiplication is very similar to the process of a 2 x 2 matrix multiplication. Again, why don't we do a matrix multiplication example?
Question 4:Find $A \bullet B$ if
First, notice that multiplying them should give us another 3 x 3 matrix. In other words,
Now let's label all our rows in the first matrix and columns in the second matrix.
Notice that the first entry of the matrix is located in the $1^{st}$ row and $1^{st}$ column, so we take the dot product of $r_{1}$ and $c_{1}$. This gives us:
Now notice that the second entry of the matrix is located in the $1^{st}$ row and $2^{nd}$ column. Thus, we take the dot product of $r_{1}$ and $c_{2}$. This gives us:
If we are to keep locating all the entries and doing the dot product corresponding to the rows and columns, then we get the final result.
We are done! Notice that the bigger the matrices are, the more tedious matrix multiplication becomes. This is because we have to deal with more and more numbers! In general, the matrix multiplication formula for 3 x 3 matrices is
## How to Multiply Matrices with Different Dimensions?
So far we have multiplied matrices with the same dimensions. In addition, we know that multiplying two matrices with the same dimension gives a matrix of the same dimensions. But what happens if we multiply a matrix with different dimensions? How would we know the dimensions of the computed matrix? First, we need to see multiplying the matrices gives you a defined matrix.
## Is the Matrix Defined?
There are cases where it is not possible to multiply two matrices together. For those cases, we call the matrix to be undefined. How can we tell if they are undefined?
The product of two matrices is only defined if the number of columns in the first matrix is equal to the number of rows of the second matrix.
Let's try to use this definition in this example.
Question 5: Let
Is $A \bullet B$ defined?
First, notice that the first matrix has 3 columns. Also, the second matrix has 3 rows. Since they are both equal to 3, then I know that $A \bullet B$ is defined.
Now that we know it is defined, how would we know the dimensions of $A \bullet B$?
## The Dimension Property
To find the dimensions of $A \bullet B$, we need to first look at the dimensions of and separately.
Now we are going to put the dimensions of the matrices side by side like this:
What we are going to do now is take the first number and the last number and combine it to get the dimensions of $A \bullet B$. See that the first number is 2 and the last number is 4. So the dimensions of $A \bullet B$ will be:
Now that we know the dimensions of the matrix, we can just compute each entry by using the dot products. This will give us:
Now that we know how to multiply matrices very well, why don't take a look at some matrix multiplication rules?
## Matrix Multiplication Properties
So what type of properties does matrix multiplication actually have? First, let's formally define everything.
Let $X, Y, Z$ be matrices, $I_{n}$ be an identity matrix, and $O_{n}$ be a zero matrix. If all five of these matrices have equal dimensions, then we will have the following matrix to matrix multiplication properties:
The associative property states that the order in which you multiply does not matter. In other words, computing $X \bullet Y$ and then multiply with $Z$ would give you the same result as computing $Y \bullet Z$ and then multiplying with $X$. Let's do an example.
Question 6: Show that the associative property works with these matrices:
Looking at the left side of the equation in the associative property, we see that $(XY)Z$ gives:
Now looking at the right side of the equation in the associative property, we see that $X(YZ)$ gives:
See how the left side and right side of the equation are both equal. Hence, we know that the associative property actually works! Again, this means that matrix multiplication order does not matter!
Now the next property is the distributive property. The distributive property states that:
We see that we are allowed to use the foil technique for matrices as well. Just to show that this property works, let's do an example.
Question 7: Show that the distributive property works for the following matrices:
See that the left hand side of the equation is $X(Y + Z)$. Hence computing that gives us:
Now let's check if the right hand side of the equation gives us the exact same thing. Notice that the right hand side of the equation is $XY + XZ$. Computing this gives us:
Notice that the left hand side of the equation is exactly the same as the right hand side of the equation. Hence, we can confirm that the distributive property actually works.
## Is Matrix Multiplication Commutative?
We know that matrix multiplication satisfies both associative and distributive properties, however we did not talk about the commutative property at all. Does that mean matrix multiplication does not satisfy it? It actually does not, and we can check it with an example.
Question 8: If matrix multiplication is commutative, then the following must be true:
Show that $XY eq YX$ if
First we compute the left hand side of the equation. Calculating $XY$ gives us:
Now computing the right hand side of the equation, we have:
As you can see,
Because we have
These two matrices are completely different.
Now there are still a few more properties of the multiplication of matrices. However, these properties deal with the zero and identity matrices.
## Matrix Multiplication for the Zero Matrix
The matrix multiplication property for the zero matrix states the following:
where $O$ is a zero matrix.
This is means that if you were to multiply a zero matrix with another non-zero matrix, then you will get a zero matrix. Let's test if this is true with an example.
Question 9: Show that the equation $OX = O$ and $XO = O$ holds if:
Let's first look at the equation
Notice that calculating $OX$ gives us:
We do see that $OX = O$, so the equation holds. Similarly, if we calculate $XO$, we get:
We do see that the equation $XO = O$ holds, so we are done.
## Matrix Multiplication for the Identity Matrix
Now what about the matrix multiplication property for identity matrices? Well, the property states the following:
where $I_{n}$ is an $n \times n$ identity matrix. Again, we can see that the following equations do hold with an example.
Question 10: Show that the equations $X I_{2} = X$ and $I_{2} X = X$ holds with the following matrices
So for the equation $X I_{2} = X$, we have:
So the equation does hold. Similar to the equation $I_{2}X = X$, we have:
Again, the equation holds. So we are done with the question, and both equations hold. This concludes all the properties of matrix multiplication. Now if you want to look at a real life application of matrix multiplication, then I recommend you look at this article.
https://www.mathsisfun.com/algebra/matrix-multiplying.html
In this section we will learn how to multiply two matrices like A $\cdot$ B together.
A $n$-tuple is an ordered list of $n$ numbers. For example,
(1,2,3,4) is an ordered quadruple with 4 numbers, and (1,2,3) is an ordered triple with 3 numbers. We usually specify each ordered $n$-tuple as a variable with an arrow on top. For example,
$\vec{x}=(1,2,3,4)$
If we have 2 ordered $n$-tuples, then we can find the dot product. The dot product is summation of all the product of each corresponding entries. For example, let
$\vec{a}=(1,2,3)$ and $\vec{b}=(2,2,2)$.
If we do the dot product of $\vec{a}$ and $\vec{b}$ and ,then we will get the following:
$\vec{a}\cdot\vec{b}=(1,2,3)\cdot(2,2,2)$
$=1\cdot2+2\cdot2+3\cdot2$
$=2+4+6$
$=12$
When we want to multiply a matrix by a matrix, we want to think of each row and column as a $n$-tuple. To be exact, we want to focus on the rows of the first matrix and focus on columns of the second matrix. For example,
For example, $\vec{r_1}$is the first row of the matrix with an ordered triple (1,2,3). Now to multiply these two matrices, we need to use the dot product of $\vec{r_1}$ to each column, dot product of $\vec{r_2}$ to each column, and $\vec{r_3}$ to each column. In other words<|endoftext|>
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A preposition of place is a word which helps to describe where something or somebody is. A preposition of place describes the location of something.
There are many prepositions of place in the English language and each of them has several uses.
In this lesson, we will study 3 prepositions of place:
Specific point or place
We can use “at” to describe a very specific point or place. Examples:
There is a dog at the top of the steps.
There is a woman at the bottom of the steps.
The postman is at the door.
There is a supermarket at the end of the street.
An exact address
An exact address is an address with the name of the street and also the number of the street. Example:
Mark lives at 55 Oxford Street.
If we do not know the exact address. If we only know the name of the street, then we use “on” as the preposition:
Mark lives on Oxford Street.
We use the preposition “at” to describe being present at an event. Examples:
I was at a party last night.
Where is David?
He is at a concert.
Buildings where an event or activity takes place
We use “at” if somebody has gone to a building, and we want to describe generally and roughly where the person is. When we use “at” in this context it is not important if the person is currently inside the building or outside the building. We are simply describing where the person has gone. It is not specific. Examples:
Mark: Hello David. Is Jane here?
David: Jane isn’t here. She’s at the library with her friends.
David is saying that Jane has gone to the library. It is a general statement of where Jane is. Maybe Jane is inside the library building or maybe she is outside the library building waiting. He doesn’t know and it isn’t important.
If we want to be very specific about a building, then we use the preposition of place “in” to say that the person has entered the building. Example:
(Mark goes to the library to look for Jane. He sees a friend of Jane’s called Sarah in front of the library .)
Mark: Hello Sarah. Have you seen Jane? Where is she?
Sarah: Hi Mark. Yes, she is in the library.
Conclusion, for buildings, we use “at” for a general description of where somebody has gone and we use “in” to specifically say the person is inside the building.
A stop on a journey
We stopped at a small village.
The train from Manchester to London stops at Birmingham.
Enclosed space / a large place with boundaries
We use the preposition “in” for an enclosed space or a place which is surrounded by boundaries. Examples:
The dog is in the garden.
I have an apple in my bag.
David’s car is in the car park.
Let’s go for a walk in the forest.
Towns or cities
A very common use of the preposition “in” is for towns and cities. In english, we do NOT use “at” for towns and cities:
I was born in Manchester.
Jane lives in London.
Jane lives at London.
Where is Mark?
He’s in Birmingham today.
For a surface
The preposition “on” is often used to describe a surface. Examples:
There’s a clock on the wall.
What’s that on the ceiling?
There’s a bag on the floor.
My books are on the table.
If something is physically attached or joined to something else, then we use the preposition “on”. Example:
She is wearing a ring on her finger.
Close to a river
If something is directly next to a river, then we use the preposition “on”:
London is on the River Thames.
My house is on the River Avon.<|endoftext|>
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Everyone engages in economic behavior, but the principles behind those behaviors are generally not known. This leads to harmful decisions, both at an individual and a social level. This course serves to provide students a rudimentary introduction to basic economic principles in an effort to help them make more sense of the world around them. It is hoped that what is learned in this course will help inform their decision-making as they grow in age and wisdom. Being a cursory introduction, this course would serve to provide a semester-long course.
- 3 Course Sections: the first introduces the concepts, the second explores how the concepts work together, and the third explores how those concepts are put into practice
- 30 Video Lessons
- About 5 hours of instructional videos, and over 3 hours of review videos
- Lessons include contemporary examples, as well as examples from the past, enriching students' understanding of economics in history
- 30 Enrichment activities/exercises to help explore lesson content further
- 30 Video reviews, which review the Enrichment Activities
- 3 end-of-section assessments, with answer keys
In Ancient Greece and Rome, students were exposed to a three-tiered approach. First, students learned "vocabulary", which included all the important terms and concepts of a discipline. After the vocabulary was mastered, the students explored "grammar", which included the rules of how to use the concepts earlier studies. Finally, students explored "rhetoric", the art of implementing the rules. Rhetoric was never finite, and encouraged the student to grow as they learned.
This course follows the same format:
Section 1 defines key principles in the study of economics
Section 2 looks at how those principles work, and how they are related to other concepts
Section 3 explores 5 different theories on how to apply the economic concepts discussed.
This course is oriented toward all students, grades 10-12, interested in a cursory understanding of economics.
By the end of this course, students will be able to simply define key economic principles, like supply, demand, price, value, and similar terms. They will also be able to apply various principles to situations they find themselves in. Further, they will be able to identify different theoretical approaches to applying those principles, and evaluate those approaches. Because the course draws from history, the student will also be able to explain various historical events related to economics. In the process of this course, students will develop and refine research skills.
Being an introduction, but one that uses historical examples, the prospective student should have at the least studied US History a little.
Section 1: Principles
What is Economics?--defines economics
Scarcity--defines the concept of scarcity
Efficiency and Effectiveness--defines these sister concepts, and looks at how they relate to scarcity
Resources--defines resources, and looks at different types of resources
Cost, Price, and Value--defines and differentiates between these related concepts
Supply and Demand--defines these mirror concepts
Markets--looks at what constitutes a "market" and introduces the "circular flow" diagram
Point of Diminishing Returns--looks at a concept often neglected, but important
Money--defines money, and looks, briefly, at how it came about
Section 2: Mechanics
Money Supply--looks at how money supply is manipulated
Price--looks into where the prices you pay come from
Demand and Price--looks at the relationship between demand and price
Supply and Price--looks at the relationship between supply and price
Money Supply and Price--looks into the relationship between money supply and price
Inflation and Deflation--explores the terms inflation and deflation
Price Corrections--explores the concept of "price corrections" and their impact on peopel
Economic Cycle--looks at the cyclic nature of economics
Economic Indicators--introduces and explains some markers called indicators
Competition vs Monopoly--compares the opposing economic concepts of competition and monopoly
Competition--looks at how competition is supposed to work
Monopoly--looks at various ways monopolies are created, and their impacts
Modern Markets--builds on the basic market explored in Section 1
Banks--explores the role fo banks in economics
Government--explores the role of government in economics
Labor and Wages--looks at the need for labor, and the wages labor requires
Section 3: Theories
The Free Market--explores the first market-based theory
Capitalism--looks into the second market-based theory
Social Capitalism--introduces a popular variation of market-based theory
Socialism--introduces and summarizes the initial alternative of market economics
Communism--introduces and summarizes the second most influential alternative to market economics<|endoftext|>
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# How do you find the exact value of arctan((arctan(9/7) - arctan(7/6)) / (arctan(5/3) - arctan(3/2)))?
Sep 9, 2015
Find arctan
#### Explanation:
$\tan x = \frac{9}{7}$ --> arc x = 52.15 deg
$\tan y = \frac{7}{6}$ --> arc y = 49.39 deg
arc x - arc y = 52.19 - 49.39 = 2.73 deg
$\tan u = \frac{5}{3}$ --> arc u = 59.03 deg
$\tan v = \frac{3}{2}$ --> arc v = 56.30 deg
arc v - arc u = 59.03 - 56.30 = 2.73
$\frac{a r c x - a r c y}{a r c u - a r c v} = \frac{2.73}{2.73} = 1$.
$\tan A = \frac{\arctan \left(\frac{9}{7}\right) - \arctan \left(\frac{7}{6}\right)}{\arctan \left(\frac{5}{3}\right) - \arctan \left(\frac{3}{2}\right)} = 1$--> $a r c A = \frac{\pi}{4}$<|endoftext|>
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Color theory is a vast and complicated sphere of knowledge. It consists of different scientific elements, such as: optics, spectroscopy, human anatomy and physiology, psychology, art history and theory, philosophy, ethics, architecture theory, design and many other applied sciences. In this article we will show only the schemes of harmonic color combinations and the examples of their usage by a bunch of talented vector artists.
How are Colors Created?
Let's first understand how the varieties of all colors are formed. All the colors can be received from the combination of primary colors, which are red, yellow and blue. These colors differ from the others by the fact that they cannot be created by mixing the other colors.
In order to make secondary colors you should mix the primary ones. By mixing red with yellow we get orange, and by mixing red with blue you get a violet color. Blue together with yellow turns into green.
For creating tertiary colors you should mix the primary color with the neighboring secondary color. It means that there are six tertiary colors (two colors from each primary color). As a rule colors are placed in a spectrum and it is called a color wheel.
The method of making your own color palette with the help of Color Blending Methods is described in a wonderful tutorial How to Create a Wide Range of Custom Color Swatches in Illustrator. One of the methods is illustrated in the picture below. You can use these colors in your work after downloading the source file.
The figures in the picture show that 1 - is for primary colors, 2 - is for secondary and 3 - is for tertiary colors. Colors that have figures are called the pure colors; they have no black and white impurity. For getting colors in digital art such color mode as RGB, CMYK, LAB and HSB can be used. You can read everything about their quantities and application in the article An Introduction to Illustrator's Color Tools.
The Color Scheme
There are different methods of getting harmonic color combinations. Let's discuss harmonic color schemes and examples of their application in vector art. These color schemes are usually called the basic ones.
Monochromatic Color Scheme
The variations of brightness and intensity of one color is used in this scheme. This scheme is simple and elegant, colors are soothing. The basic colors can be combined with neutral ones such as white, black and gray to contrast the elements of a composition.
Golden Time, by Guilherme Marconi
The Curse of the Hamster, by Zutto
Analogous Color Scheme
In this scheme we use colors that are placed near each other in the color spectrum. This kind of scheme is often used for the creation of peaceful and comfortable designs. An analogous color scheme is often met in nature.
One color is usually used as the basic one, the second color is as accompanied, the third is used as an accent for the contrast creation, it may be white, black or gray.
Bring Peace to Midnight, by lostsoulx44
phil's_lion, by Melelel
Complementary Color Scheme
Colors that are opposite each other in a color spectrum are called complementary colors (for example red and green).
Such color combinations creates a high level of contrast in combination. But we should be careful with such a combination and use it only if we want to select something. Complementary colors are really bad for text.
Fruits, by Konstantin Shalev
TULIP, by gartier
Split Complementary Color Scheme
This scheme is the variation of complementary color scheme. It uses a color and two colors adjacent to its complementary. Only right or left colors from a complementary color are used. This provides high contrast without the strong tension of the complementary scheme. The split complementary scheme is harder to balance than monochromatic and analogous color scheme. One warm concentrated color and a number of cold colors are usually used.
The Daughter of Poseidon, by zanthia
OJ, by Ikue
Triadic Color Scheme
Three colors that are equally placed in a color spectrum are used in this scheme. This scheme gives a strong visual contrast with harmony and color richness. Colors in this scheme are more balanced than in a complementary color scheme. As a rule one color in a composition is chosen as basic.
Bananas-and-Tomatoes, by rosesaregreen
Summer, by Zzanthia
Tetradic (Double Complementary) Color Scheme
This scheme is the richest among all the schemes because four colors which are placed in complementary pairs are used. It is difficult to achieve harmony in this scheme. If four colors are used in the same quantity the composition might appear to be unbalanced. That's why we should choose one color which will be dominant.
Time, by LimKis
Fruit Fly, by ArtSerenity
Adobe Illustrator allows you to use basic color schemes. To do this, after selecting the color you should open the tab Harmony Rules in the Color Guide palette. Select the desired color scheme and choose color from a number of proposed colors.
Except the basic colors, there are lots of other color schemes that are based on associations with human reception. These are the main schemes.
The tones of red are considered to be hot, which are associated with the fire. Hot colors seem to move beyond the composition flatness and attract attention. That's why they are often used in posters, ads, road signs. Hot colors possess the strength and aggressiveness.
Nissan Skyline GT-R Vector, by hoshiboshi
Tones of blue are considered to be cold colors. Cold colors remind us of ice and snow. When cold and hot colors are placed nearby it seems as if they are vibrating.
Lost in the Space, by javieralcalde
Life, by Surround
All colors that contain red seem to be warm. Red-orange, orange and yellow-orange are supposed to be warm colors. Warm colors are comfortable, impulsive and friendly.
Autumn Girl, by mashi
Sunset Room, by SteveNewport
The basic tone of cool colors is blue. If we add yellow to cool colors we receive yellow-green, green and green-blue colors. Such tones relax, refresh, and give a feeling of depth and comfort.
Frozen in Time, by hitman101
The Martini Drink - Gmesh, by enikOne
These colors have got lots of white color. As a rule such colors are called pastels. It seems that they are transparent and weightless. The more light the color is, the less is the number of combination variations with it. The light colors are open and give the feeling of reconciliation and quietness.
Glossy Roseate, by afordite
I'll wait for you..., by NaBHaN
The dark colors are those colors that are mixed with black. They look as if they close the space, make it less. Dark colors are associated with autumn and winter, which are dark seasons. The combination of light and dark colors creates the feeling of drama.
Pirate, by LimKis
Dark elf with BFS, by sygnin
The brightness of color is measured by the quantity of pure color. The color brightness is made in the absence of a black color. Bright colors attract our attention, that's why it is used in packaging and ads.
Study and Fun, by mrbumbz
Other, by LimKis
Today we have only opened the door to the complicated and wonderful universe of colors. The feeling of harmony and its application can be learned, as well as the science, and principles behind it. I wish you success in pursuing each fascinating color world.
More Color Theory and Color Harmony Information
Learn more from these resources:<|endoftext|>
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# Level 4/5 Booster Lesson 1B Place value.
## Presentation on theme: "Level 4/5 Booster Lesson 1B Place value."— Presentation transcript:
Level 4/5 Booster Lesson 1B Place value
Place value decimal point digit Objectives:
Understand and use decimal place value. Multiply and divide by 10 and 100 and explain the effect. Vocabulary Place value decimal point digit
35 350 3500 H T U Multiply this number by 10 What is your answer?
What has happened to all of the digits? 3 tens become 3 hundreds, 5 units become 5 tens Each digit in the number has moved one column to the left. Th H T U What happens when we multiply by 100? 3500 (each digit in the number moves 2 columns to the left)
Multiply each of the following numbers by 10, as they appear.
40 23 123 56 230 560 1230 400 Multiply each of the following numbers by 100, as they appear. 340 35 213 20 3500 2000 34 000 21300
46 4.6 0.46 . t T U U . t U . t h Divide this number by 10
What is your answer? U t 4.6 Explain what has happened to each of the digits. The 4 tens become 4 units, the 6 units become 6 tenths. Each of the digits in the number has moved one column to the right. U t h What happens when we divide by 100? 0.46 Each of the digits in the number moves two columns to the right.
Divide each of the following numbers by 10, as they appear.
57 70 350 134 5.7 13.4 35.0 7.0 Divide each of the following numbers by 100, as they appear. 409 450 50 39 4.50 4.09 0.50 3.9
T1.1B As the number is highlighted multiply or divide by powers of 10 as instructed. 400 3 30 500 1000 70 9 40 20 50 200 8 300 5 60 700
30 3 0.3 3000 30000 300 3 x 1 = 3 3 x 10 = 30 X ? X ? X ? 3 x 0.1 = 0.3 3 x 1000= 3000 X ? X ? X ? 3 x 100 = 300 3 x = Look carefully at this spider diagram before you move on.
Multiplication Spider Diagram
W/S1.1B Using worksheet W/S 1.1B complete your own multiplication spider diagram. Choose your own number in the middle. Now check your answers using a calculator.
Try to state the answer before it appears on the spider diagram.
4.65 ÷ 100 ÷ 10 46.5 465 ÷ 1 000 465 ÷ 1 0.465
Now check your answers on a calculator before you move on.
Division Spider Diagram W/S 1.2B Start with a number of your choice in the middle and complete the division spider diagram. Now check your answers on a calculator before you move on.
T1.2B 0.3 30 6 40 400 5 0.6 0.5 50 4 300 0.4 3 500 60 600 I divided by ten and my answer is 0.6 – what number did I start with? I multiplied by 100 and my answer is 40 – what number did I start with? Make up three questions like the ones above and try them out on someone else – explain your working to them if necessary.
Place value decimal point digit Objectives:
Understand and use decimal place value. Multiply and divide by 10 and 100 and explain the effect. Vocabulary Place value decimal point digit
Thank you for your attention
Similar presentations<|endoftext|>
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# Daily Archives: January 18, 2020
## AMS panel on math and motherhood
To top off JMM, I went to the AMS panel discussion on mathematics and motherhood hosted by Carrie Diaz Eaton of Bates College. Panelists Karoline Pershell, executive director of the Association for Women in Mathematics, Karen Saxe, the associate executive director of the AMS, and Talithia Williams, an associate dean and professor at Harvey Mudd, discussed everything from how having children affected their careers, family leave policies and informal support networks, role models, and accepting outside help. Here are some highlights from the panel, edited for clarity:
## James Tanton’s proofs that 1 = 2
This afternoon at Mathemati-Con, James Tanton proved to us 12 different ways that 1 = 2. Here are just a few of his arguments:
1. Since $i^2 = -1$,
$1 = {i^2 + 3 \over 2} = {(\sqrt{-1})(\sqrt{-1}) + 3 \over 2} = {\sqrt{(-1)(-1)} + 3 \over 2} = 2$
2. Take a unit square. The diagonal has length $\sqrt{2}$ by the Pythagorean theorem. But we can also approximate the diagonal using a staircase. The lengths of the horizontal segments of the staircase must add to 1, as must the lengths of the vertical segments. So the staircase, no matter how closely it approximates the diagonal, always has length 2. As we make the approximation better, this means that $\sqrt{2} = 2$ and therefore $2 = 4$ implies $1 = 2$.
3. Write $0 = 0 + 0 + 0 + \dots$. Since $0 = 1 – 1$, we have
$0 = (1-1) + (1-1) + (1-1) + \dots$
But if we rearrange the parentheses, we get
$0 = 1 + (-1+1) + (-1+1) + (-1 +1) + \dots = 1 + 0 + 0+ \dots = 1.$
Thus $0 = 1$; adding 1 to both sides gives $1 = 2$.
See if you can spot the mistakes in each of the arguments. Tanton ended with the following irrefutable “proof by shopping”: At a store holding a 2-for-1 sale, he asked the sales associate how much it would cost to buy 1 item. She responded that it was the regular price–that is, the same price as buying 2 items. Hence it must follow that $1 = 2$.
## Advice from Haimo Award Winners
What do award-winning mathematicians Federico Ardila-Mantilla, Mark Tomforde, and Suzanne Weeks have to say about teaching? Each of these mathematicians received the MAA Haimo Award at JMM 2020 and gave 20 minute talks. Here is a summary.
Federico Ardila-Mantilla used his time to ask us to reflect by thinking about three “open questions.” First, a story about Federico. A mathematician during the day, DJ at night, it is not surprising that Federico is interested in music. This interest led him to enroll in a timbales class at his local community center. His teacher? One of the top timbaleros of the 1970’s. After just two weeks of classes, this timbalero invited Federico to play with him at an event. It was a moving experience for Federico. What does a similar experience look like in a math environment? When do we see top mathematicians inviting newcomers into their work?
Undisturbed beech forest behave in a weird but spectacular way. “Apparently, the trees synchronize their performance so that they are all equally successful. And that is not what one would expect. Each beech tree grows in a unique location, and conditions can vary greatly in just a few yards. The soil can be stony or loose. It can retain a great deal of water or almost no water. It can be full of nutrients or extremely barren. Accordingly, each tree experiences different growing conditions; therefore, each tree grows more quickly or more slowly and produces more or less sugar or wood, and thus you would expect every tree to be photosynthesizing at a different rate… [However] The rate of photosynthesis is the same for all the trees. The trees, it seems, are equalizing differences between the strong and the weak.” – The Hidden Life of Trees. What does this synchrony look like in a mathematics classroom?
Born in 1952, bell hooks experiences the initial racial integration after segregation was deemed illegal. “Bussed to white schools, we soon learned that obedience, and not a zealous will to learn, was expected of us. That shift taught me the difference between education as the practice of freedom and education that merely strives to reinforce domination.” – bell hooks. What does a liberating math classroom look like?
Mark Tomforde had four original beliefs that have changed throughout his teaching experiences. They are:
– Old Belief 1: My main objective is to teach math content. The more content the better. Updated belief: My main objective is to teach my students to think. In 10 years what will students remember from this course? What do I want them to remember? I want to prepare my students for life, to develop good habits when engaging with facts, to work on improving their skills such as problem solving, attention to details, ability to communicate technical ideas, and to think deeply about a few topics.
– Old Belief 2: The more students I can reach the better. Updated belief: Quality is more important than quantity. I am a fixed quantity and the more projects I am involved with, the less of myself I can devote to them. For me, it is better to devote my time and energy to one or two projects and do it well rather than spending little time in a lot of projects.
– Old Belief 3: The best students are our future, teach to them. Updated belief: Direct teaching to all students paying particular attention to the ones that are struggling. Are our best students more important? Sometimes we think we are just training the future mathematicians and try and identify the “best students.” But, how many students are we not able to reach? Our goal should not be to find talent but rather to help students develop their talent.
– Old Belief 4: Good students are the ones who can prove theorems, solve problems and do well. Updated belief: There are many ways for students to be successful and contribute. When I teach math majors, I regard all of my students as the future of math. This is because math requires a much larger ecosystem than a group of researchers. In addition to researchers, we need teachers, advocates, PRs, writers, and many others are needed. In my classes, I ask students to find a need and contribute to fill that need. Some of the projects students conduct are: leading math circles, tutoring, creating guides for incoming students in the department, starting newsletters, and organizing department picnics.
Suzanne Weeks is a co-director of MSRI-UP and PIC Math, and has worked in many industrial projects. Her experiences have taught her it’s important to:
Show up – Take advantage of the opportunities that arise for you.
Sit in front – Be engaged, be involved, be fully in the room.
Raise your hand – Be an active participant, raise your hand, raise your voice, ask for stuff when you need it.
Make friends – You need to have a support group.
Be yourself – Your path is your own. Do what is best for you.
– Know your worth because you are worthy – Celebrate your successes.<|endoftext|>
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How do you bring an endangered plant species back from the brink of extinction? The answer might be found in zoo animals.
That’s the inspiration for Chicago Botanic Garden scientist Jeremie Fant’s latest research. Fant, a molecular ecologist and plant genetics guru, is working with other botanic gardens around the world to develop conservation and reintroduction plans modeled after the ones used by zoos to protect endangered animal species.
“When we conserve plant species, it’s possible to preserve hundreds of individuals, and the genetic information they contain, by banking their seed or using cuttings to propagate them,” said Fant. “But when this is not possible, these plant collections are maintained by continually crossing with other plants to produce new seed. This is akin to animals in zoo collections. Zoos have used genetic information to develop ‘studbooks’ to decide what crosses are compatible so they maintain genetic diversity and prevent inbreeding.”
Fant’s work is based on zoological cases including black-footed ferrets in the 1980s. Zoologists created a breeding program that ultimately reintroduced the threatened species back into the wild. The zoologists used genetic information taken from the remaining black-footed ferrets, and bred a strong, biodiverse population that could keep the animals healthy and, more importantly, increase numbers, which is the aim of all good conservation programs.
Fant’s work centers on one plant in particular: the Brighamia insignis, or “Cabbage on a stick,” or as we’ve fondly named it, “Cabby.” This is Cabby’s story:
To stay tuned on what Fant, and the rest of the Garden’s conservation scientists are doing, check out the latest news at chicagobotanic.org/research.
Competition is heating up in the western United States. Invasive and native plants are racing to claim available land and resources. Alicia Foxx, who studies the interplay of roots of native and invasive plants, is glued to the action. The results of this contest, says the plant biology and conservation doctoral student at the Chicago Botanic Garden and Northwestern University, could be difficult to reverse.
Cheatgrass, which is an aggressive, invasive plant with a dense root system, is in the lead and spreading quickly across the west. Native plants are falling in its wake—especially when it comes to their delicate seedlings that lead to new generations.
Foxx is one of the scientists working to give native plants a leg (or root) up. She hypothesizes that a carefully assembled team of native plant seedlings with just the right root traits may be able to work together to outpace their competition.
“We often evaluate plants for the way they look above ground, but I think we have to look below ground as well,” she said. Foxx’s master thesis focused on a native grass known as squirreltail, and her hypothesis addressed the idea that the more robust the root system was in a native grass, the better it was at competing with cheatgrass. Now, “I’m looking more at how native plants behave in a community, as opposed to evaluating them one by one… How they interact with one another and how that might influence their performance or establishment in the Colorado plateau.”
In the desert climate, human-related disturbances such as mining, gas exploration, livestock trampling, or unnaturally frequent fires have killed off native plants and left barren patches of land behind that are susceptible to the arrival of cheatgrass.
“Some of our activities are exacerbating the conditions [that are favorable for invasive plants]. We need to make sure that we have forage for the wildlife and the plants themselves, because they are important to us for different reasons, including the prevention of mudslides,” she said. “We are definitely confronted with a changing climate and it would be really difficult for us to reverse any damage we have caused, so we’re trying to shift the plant community so it can be here in 50 years.”
Garden conservation scientist Andrea Kramer, Ph.D. advises Foxx, and her mentorship has allowed Foxx to see how science theories created in a laboratory become real-life solutions in the field. “I think I’m very fortunate to work with Andrea, who works very closely with the Bureau of Land Management…it’s really nice to see that this gets replicated out in the world,” said Foxx. Seeds from their joint collecting trip in 2012 have been added to the Garden’s Dixon National Tallgrass Prairie Seed Bank.
In a way, Foxx is also learning from the invasive plants themselves. To develop her hypothesis, she considered the qualities of the invasive plants; those that succeeded had roots that are highly competitive for resources. After securing seeds from multiple sources, she is now working in the Garden’s greenhouse and the Population Biology Laboratory to grow native plants that may be up to the challenge. She is growing the seedlings in three different categories: a single plant, a group of the same species together, and a group of species that look different (such as a grass and a wildflower). In total, there will be 600 tubes holding plants. She will then evaluate their ability to establish themselves in a location and to survive over time.
There has been very little research on plant roots, but Foxx said the traits of roots, such as how fibrous they are, their length, or the number of hair-like branches they form, tell us a lot about how they function.
“I’m hoping that looking at some of these root traits and looking at how these plants interact with one another will reveal something new or solidify some of the theories,” said Foxx.
She aims to have what she learns about the ecology of roots benefit restorations in the western United States. It is possible that her findings will shape thoughts in other regions as well, such as the prairies of the Midwest. Future research using the seeds Foxx collected could contribute to the National Seed Strategy for Rehabilitation and Restoration, of which the Garden is a key resource for research and seeds for future restoration needs.
The Chicago native has come a long way since she first discovered her love of botany during high school. After completing her research and her Ph.D., she hopes to nurture future scientists and citizen scientists through her ongoing work, and help them make the connections that can lead to a love of plants.
The National Parks provide dream vacations for us nature lovers, but did you know they also serve as vital locations for forward-thinking conservation research by Chicago Botanic Garden scientists?
From sand to sea, the parks are a celebration of America’s diversity of plants, animals, and fungi, according to the Garden’s Chief Scientist Greg Mueller, Ph.D., who has worked in several parks throughout his career.
“National Parks were usually selected because they are areas of important biodiversity,” Dr. Mueller explained, “and they’ve been appropriately managed and looked after for up to 100 years. Often times they are the best place to do our work.”
As we celebrate this centennial year, he and his colleagues share recent and favorite work experiences with the parks.
Take a glimpse into the wilderness from their eyes.
This summer, Mueller made a routine visit to Indiana Dunes National Lakeshore to examine the impact of pollution and other human-caused disturbances on the sensitive mushroom species and communities associated with trees. “One of the foci of our whole research program (at the Garden) is looking at that juxtaposition of humans and nature and how that can coexist. The Dunes National Lakeshore is just a great place to do that,” he explained, as it is unusually close to roads and industry.
Evelyn Williams, Ph.D., adjunct conservation scientist, relied on her fieldwork in Guadalupe Mountains National Park to study one of only two known populations of Lepidospartum burgessii, a rare gypsophile shrub, during a postdoctoral research appointment at the Garden. “We were able to work with park staff to study the species and make recommendations for management,” she said.
As a Conservation Land Management intern, Coleman Minney surveyed for the federally endangered Ptilimnium nodosum at the Chesapeake and Ohio Canal National Historical Park earlier this year. “The continued monitoring of this plant is important because its habitat is very susceptible to invasion from non-native plants,” explained Minney, who found the first natural population of the species on the main stem of the Potomac River in 20 years.
According to conservation scientist Andrea Kramer, Ph.D., “In many cases, National Parks provide the best and most intact examples of native plant communities in the country, and by studying them we can learn more about how to restore damaged or destroyed plant communities to support the people and wildlife that depend upon them.”
The parks have been a critical site for her work throughout her career. Initially, “I relied on the parks as sites for fieldwork on how wildflowers adapt to their local environment.”
Today, she is evaluating the results of restoration at sites in the Colorado Plateau by looking at data provided by collaborators. Her data covers areas that include Grand Canyon National Park, Capitol Reef National Park, and Canyon de Chelly National Monument.
Along with colleague Nora Talkington, a recent master’s degree graduate from the Garden’s program in plant biology and conservation who is now a botanist for the Navajo Nation, Dr. Kramer expects the results will inform future restoration work.
At Wrangell–St. Elias National Park and Preserve in Alaska, Natalie Balkam, a Conservation Land Management intern, has been hard at work collecting data on vegetation in the park and learning more about the intersection of people, science, and nature. “My time with the National Park Service has exposed me to the vastly interesting and complex mechanics of preserving and protecting a natural space,” she said. “And I get to work in one of the most beautiful places in the world—Alaska!”
The benefits of conducting research with the National Parks extend beyond the ability to gather high-quality information, said Mueller. Parks retain records of research underway by others and facilitate collaborations between scientists. They may also provide previous research records to enhance a specific project. Their connections to research are tight. But nothing is as important as their ability to connect people with nature, said Mueller. “That need for experiencing nature, experiencing wilderness is something that’s critical for humankind.”
For research and recreation, we look forward to the next 100 years.
Golden paintbrush (Castilleja levisecta) is gaining ground in its native Oregon for the first time in more than 80 years. Recent reintroductions have seen the charismatic species flourish on its historic prairie landscape. To keep the momentum going, scientists are pulling out all the stops to ensure that the new populations are robust enough to endure.
“Genetic variability will be key to the reintroduction success of golden paintbrush,” explained Adrienne Basey, graduate student in the plant biology and conservation program of the Chicago Botanic Garden and Northwestern University.
Basey, who previously managed a native plant nursery, is now studying the genetic diversity of golden paintbrush plants before, during, and after they are grown in a nursery prior to reintroduction to the wild.
“My work is looking at the DNA, or genetics, of the wild, nursery, and reintroduction populations to see if there is any change through that process,” she said. If there is a change, she will develop recommendations for adjusting the selection and growing process to better preserve diversity. “My goal is to give both researchers and practitioners more information to work with,” she noted.
Building for the Future
The research is unique in the relatively young field of restoration science, according to Basey’s co-advisor and molecular ecologist at the Garden, Jeremie Fant, Ph.D. “Adrienne’s study is awesome because of the fact that it has data and the samples to back it up; it is early on in this game of reintroductions and restorations, and potentially could have a lot of impact, not just for that species but what we tell nurseries in the future,” he said.
Basey is working with data collected over the past decade by research scientists at the Institute for Applied Ecology in Corvallis, Oregon, and University of Washington herbarium specimens from Washington and Oregon dating as far back as the 1890s, and data she has collected from existing plants during field work. “It’s a perfect partnership,” said Dr. Fant, who noted that the Garden is guiding the molecular aspect of the study while colleagues in Washington and Oregon are providing a large portion of the data and samples.
The availability of all of this information on a single species that is undergoing restoration is very rare, explained Fant. “It’s a very unique scenario that she has there, so we can look at how diversity changes as we go from step to step and hopefully identify any potential issues and where they are occurring in the process.”
The study itself will likely serve as a research model for other species in the future. “There isn’t much research out there to help propagators understand when and where genetic diversity may be lost during the production process,” said Basey’s co-advisor and conservation scientist at the Garden, Andrea Kramer, Ph.D.
Last year, Basey, Fant, and Kramer worked together to write a paper outlining ten rules to maximize and maintain genetic diversity in nursery settings. “My goal is to support reintroduction efforts by informing nursery practices and demonstrate to nurseries on a broader scale how their practices can influence genetic diversity through a single case study,” said Basey.
A Green Light Ahead
Her preliminary research is focused on four golden paintbrush populations. Early evaluations show clear distinctions between a few of them, which is good news. Basey will next compare those genetic patterns to those of plants in reintroduction sites.
According to Fant, earlier studies by other researchers have shown that many restoration efforts for threatened species suffer from low levels of genetic diversity prior to reintroduction, due to a number of causes ranging from a small population size at the outset to issues in propagation. It is critical to work around those issues, he explained, as the more genetic diversity maintained in a population, the better equipped it is to survive environmental changes from drought to temperature shifts.
Basey will also compare the current level of diversity of golden paintbrush to that of its historic populations, to get a better sense of what the base level should be for reintroduction success. She plans to wrap up her lab work well before her summer 2015 graduation date.
For now, she is pleased with the level of diversity she sees in the current population. “I think the fact that it has a high genetic diversity means that these reintroductions could be successful,” she said. “But if we are creating a bottleneck, we need to know that so we can mitigate it as quickly as possible.” (A bottleneck is an event that eliminates a large portion of genetic variability in a population.)
Fant is enthusiastic about the timing of the study as the field of restoration is taking off. “We can jump in early as programs are being started,” he noted. “If we all learn together, I think it really does ensure that everyone gets what they need in the end.”
For Basey, it’s about building a bridge between the theoretical and the applied aspects of restoration. “My interest isn’t so much in this single species but more in the communication of science to practitioners. I like to bridge the line between research and the people who are using research,” she said.
Basey, like the golden paintbrush, is looking toward a bright future.
Huddled on a sand dune, the small community of bristly Lepidospartum burgessii plants would be easy for most of us to overlook. But to scientists from the Chicago Botanic Garden, the rare shrubs shine like a flare in the night sky. This is one of two known locations of the species worldwide—both in New Mexico—and the center of a rather dazzling rescue mission.
Evelyn Williams, Ph.D., a Garden postdoctoral research associate, is pulling out all the stops to save the sensitive species. Commonly called Burgess’ scale broom, it has suffered from a mysterious lack of seed production since the late 1980s.
Standing about five feet tall, the silvery-green plants only grow on gypsum dunes. They possess unique characteristics that allow them to help stabilize sand dunes in the desert conditions where they live.
“I’m interested in how we can use genetics broadly to address conservation and ecological restoration questions,” said Dr. Williams. Her curiosity led her to the Garden in 2011 to join a team of genetic experts for this formidable undertaking.
The team suspects that, because the two populations of Lepidospartum burgessii are relatively small, the existing plants have interbred and are now too closely related to pollinate one another—which means they cannot produce seeds and create new plants.
Williams set out to confirm this theory, gathering plant cuttings during summer fieldwork in 2013. She hoped to grow the cuttings into full plants that she could cross-pollinate and study at the Garden. She also took samples from 320 plants back to the Garden. There, using a microsatellite technique, she recorded the genetic pattern of each plant, noting similarities and differences.
“When we have all of these different shrubs from a population, we want to use a fine genetic tool to tease apart genetic variations,” she explained. The microsatellite approach allowed her to identify genetic markers occurring in multiple plants down to the finest level of detail.
The results were encouraging. Williams found enough genetic diversity within the two populations that they should be able to cross pollen, or DNA material, and produce seeds. “Because there is diversity in these populations, we’re really hopeful that if we do a genetic rescue we can get some seeds in these two different populations,” said Williams. A genetic rescue, she explained, is when a species is revived with the addition of new genes, which normally occurs during pollination.
That day didn’t come right away, as the cuttings failed to grow in the Garden greenhouses. Accustomed to the trial and error process of scientific discovery, Williams moved on to her backup plan.
She returned to the field in October, where she personally carried pollen-filled flowers from one population of plants to another, brushing the fluffy yellow blooms against other plants that may accept their genetic material. With plants as much as one mile apart, it was a process of patience and precision.
Williams is poised for the challenge of whatever she may, or may not, find. Ultimately, she hopes to convey a successful technique to land managers who carry out the daily work of furthering the species and enriching the biodiversity of the southwestern landscape.
“I really like that as part of the Garden we can help these public agencies and use our knowledge of genetics and conservation to stabilize and increase some of these rare populations. That’s really important to me,” said Williams.
She has been intent on advancing conservation science since childhood, inspired by her aunt, an ecologist. Her interest grew into expertise as she studied the genetics of ferns while earning her Ph.D. in botany at the University of Wisconsin.
In winter at the Garden, Williams takes every opportunity to walk through the Elizabeth Hubert Malott Japanese Garden. “I like being here in the winter and seeing a side of the Garden that’s unexpected: the snow and the beautiful structures in the Malott Japanese Garden,” she said.
Perhaps it is that perspective, of looking for the unexpected, that will unlock the mystery of Lepidospartum burgessii one day soon.<|endoftext|>
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We give a concise summary of math that helps in learning algebra for beginners at the college level. What you need to know is set theory and in particular the set of real numbers. This will help you to understand some important classes of sets such as groups, rings, polynomial rings, and fields. Also, you need to know some fundamental equalities, like remarkable identities, binomial expansion formulas, etc.
## The basics of algebra for beginners at the college level
We collect the important part of algebra to help beginners to well understand their first lectures on math at the University.
### The Basics of Sets
A set is just a collection of objects. For example, a set can be a collection of sports cars, or a set can be a collection of trigonometric functions. For example, the set that contains positive integer numbers is denoted by $\mathbb{N}:=\{0,1,2,\cdots\}$.
We say that $a$ belongs to a set $A$, and we write $a\in A,$ if $a$ is one of the collections of objects in $a$. In this case, $a$ is called an element of $A$.
We say that another set $B$ is a subset of the set $A$, and we write $A\subset B$ if all elements of $B$ are also elements of $A$. By the way, two sets $E$ and $F$ are equal if and only if $E\subset F$ and $F\subset E$.
Example: We denote by $\mathbb{R}$ the set of real numbers and by $\mathbb{Q}$ the set of rational numbers. Then $\mathbb{Q}$ is not equal to $\mathbb{Q}$. This is because $\mathbb{Q}\subset \mathbb{R}$ and $\sqrt{2}\in \mathbb{R}$, but $\sqrt{2}\notin \mathbb{Q}$. Let us give you a more advanced property of $\mathbb{R}$. We assume that the reader is familiarized with the convergence of sequences. Let $x$ be an arbitrary real number. We select the following sequence $x_n=\frac{[nx]}{n^2}$ for $n\in\{1,2,\cdots\}$, where $[nx]$ is the integer part of the real number $nx$. It verifies $nx-1<[nx]\le nx$. Thus $x-\frac{1}{n}<x_n\le x$. Hence the sequence $(x_n)_n$ converges to $x$ when $n$ goes to $+\infty$. Remark that $x_n\in\mathbb{Q}$ for any $n$. Thus any real number is a limit of a sequence of rational numbers.
The empty set denoted by $\emptyset$ is a set that contains no elements. For any set $A,$ we have $\emptyset A$. On the other hand, the subsets of $A$ form another set denoted by $\mathscr{P}(A)$. So that $$\mathscr{P}(A)=\{ B: B\subset A\}.$$ Remark that $\emptyset$ and $A$ are both elements of $\mathscr{P}(A)$.
If we have two sets $A$ and $B$ then we can define other sets like the union of $A$ and $B$ denoted by $A\cup B$, and the intersection of $A$ and $B$ denoted by $A\cap B$. The product of $A$ and $B$ defined by $$A\times B=\{(a,b):a\in A,\;b\in B\}.$$
### Map between two sets
A map $f$ between two sets $A$ and $B$ is any subset of $A\times B$. It will be denoted by $f: A\to B$ $x\mapsto f(x)$. We can define a map between $\mathbb{R}$ and the set $\{0,1\}$ bt $f(x)=1$ is $x\in [0,+\infty)$ and $f(x)=0$ if $x\in (-\infty,0)$. A map is also called a function.
A function $f: A\to B$ is called in injective if $f(x)=f(y)$ implies $x=y$. As example the function $f:\mathbb{N}\to\mathbb{}N$ defined by $f(n)=2^n$ is injective. In fact, if $f(n)=f(m)$ then $2^n=2^{m}$. Now we apply the logarithmic function, we get $n\log(2)=m\log(2)$, which implies that $n=m$.
The map $f$ is said to be surjective if, for every element of $B,$ there is at least one element $x$ in $A$ such that $y=f(x)$. The function $f(n)=2^n$ is not surjective. If not there will be $n\in \mathbb{N}$ with $3=2^n$. Thus $n=\frac{\log(3)}{2}\notin \mathbb{N},$ absurd. let $\mathbb{C}$ be the set of complex numbers. The maps $f:\mathbb{R}\times \mathbb{R}\to\mathbb{C}$ defined by $f(a,b)=a+ib$ with $i$ is the complex number satisfying $i^2=-1$. Then $f$ is surjective. In fact, we know that for any $z\in\mathbb{C}$ one can find real numbers $a\in\mathbb{R}$ and $b\in \mathbb{R}$ such that $z=a+ib=f(a,b)$.
We say that $f$ is bijective, or one-to-one if it is injective and surjective. This means that for any $y\in B,$ there is only one element $x\in A$ such that $y=f(x)$.
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## Lesson 2.5 – Regression Lines
• Make predictions using regression lines, keeping in mind the dangers of extrapolation.
• Calculate and interpret a residual.
• Interpret the slope and y intercept of a regression line.
Activity: How good are predictions using the Barbie data?
This activity connected back to the opening activity for Chapter 2. We started by telling the students that we had a hole in our data because someone in the group forgot to record the value for 5 rubber bands….so we are just going to make our best prediction. We started by just having students look at the numbers.
Of course students realized that our prediction should be between 55 and 69 cm. But then some students started to realize that each time a rubber ban was added, the lowest point seem to increase about 7 or 8 cm each time, so they wanted to add 8 to 55. Others wanted to find the midpoint between 55 and 69. This is all good thinking about linear relationships and slope.
Students than used the Applet to find the least squares regression line (they don’t know what “least squares” means until tomorrow). Then they compared their prediction to the actual value (somehow we found the lost data). Students are calculating a residual here without knowing this new vocabulary. We introduced the vocabulary at the end of the activity when we were summarizing.
The rest of the activity deals with the slope and y-intercept of the least squares regression line. This is a great algebra review for students.
The Barbie data is a great context for slope and y-intercept because they both have a very tangible physical meaning. The slope of the regression line tells us that for every additional rubber band added, we predict the distance that Barbie’s head reaches to increase by 7.646 cm. They y-intercept tells us the predicted lowest point that Barbie’s head reaches is 25.333 cm when there are 0 rubber bands. Of course this is really just a predicted value of Barbie’s height.
Notes
When we have students write out the equation for the least squares regression line, we have them (1) use context instead of x and y and (2) put a “hat” over the y variable to indicate that we are predicting y from x.
These two small changes will make predictions and residuals come much easier for students.
To help students remember the correct interpretation of slope, we took them back to Algebra 1, where they learned slope as rise/run or (change in y)/(change in x). Then we took our slope of 7.464 and wrote it as 7.464/1. When students think of slope as (change in y)/(change in x), the interpretation becomes much easier to come up with (rather than memorize!). We also made sure that students were saying “predicted lowest point” rather than just “lowest point” when interpreting slope and y-intercept.
You can preview tomorrow’s lesson by asking students how they think the Applet is figuring out which line is the “best”. You can also preview Lesson 2.7 by making students aware of the s (standard deviation of the residuals) and r2 (coefficient of determination), both of which will be interpreted later.<|endoftext|>
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Lessons | Skubes
# Search
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## Modeling Division of Fractions
00:00
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally?
## Rational Numbers : Fraction and Decimal Conversions
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Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
## Multiplying Fractions by Whole Numbers
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Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number.
Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
## Multiplying Fractions
00:00
Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get?
## Fraction
Definition: A number expressible in the form a/b where a is a whole number and b is a positive whole number. (The word fraction in these standards always refers to a non-negative number.) See also: rational number.<|endoftext|>
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# Class 12 Mathematics Chapter 4 Determinants MCQ Question with Answer
Determinants Class 12 MCQ is one of the best strategies to prepare for the CBSE Class 12 Board exam. If you want to complete a grasp concept or work on one’s score, there is no method except constant practice. Students can improve their speed and accuracy by doing more MCQ on Determinants Class 12, which will help them all through their board test.
## Determinants Class 12 MCQ Questions with Answer
Class 12 Maths MCQ with answers are given here to Chapter 4 Determinants. These MCQs are based on the latest CBSE board syllabus and relate to the latest Class 12 Mathematics syllabus. By Solving these Class 12 MCQs, you will be able to analyze all of the concepts quickly in the chapter and get ready for the Class 12 Annual exam.
Learn Determinants Class 12 MCQ with answers pdf free download according to the latest CBSE and NCERT syllabus. Students should prepare for the examination by solving CBSE Class 12 Mathematics Determinants MCQ with answers given below.
Question 1. If A, B and C are angles of a triangle, then the determinant
is equal to
(a) 0
(b) –1
(c) 1
(d) None of these
A
Question 2. If ,
then x is equal to
(a) 6
(b) ± 6
(c) – 6
(d) 0
B
Question 3.The value of
is
(a) 1
(b) 0
(c) a + b
(d) a – b
B
Question 4. The area of a triangle with vertices (–3, 0),(3, 0) and (0, k) is 9 sq. units. The value of k will be
(a) 9
(b) 3
(c) –9
(d) 6
B
Question 5. The value of determinant
(a) a3 + b3 + c3
(b) 3bc
(c) a3 + b3 + c3 – 3abc
(d) None of these
C
Question 6.If
and Aij is cofactors of aij, then value of D is given by
(a) a11 A31 + a12 A32 + a13 A33
(b) a11 A11 + a12 A21 + a13 A31
(c) a21 A11 + a22 A12 + a23 A13
(d) a11 A11 + a21 A21 + a31 A31
D
Question 7. If f(x) =
,then
(a) f(a) = 0
(b) f(b) = 0
(c) f(0) = 0
(d) f(1) = 0
C
Question 8.The value of is
(a) 1
(b) –1
(c) 0
(d) ω
C
Question 9. Let A be a square matrix of order 3 × 3, then |KA| is equal to
(a) K|A|
(b) K2|A|
(c) K3|A|
(d) 3K|A|
C
Question 10. There are two values of a which makes determinant Δ =
,86 then sum of these numbers is
(a) 4
(b) 5
(c) – 4
(d) 9
C
Whoever needs to take the CBSE Class 12 Board Exam should look at this MCQ. To the Students who will show up in CBSE Class 12 Mathematics Board Exams, It is suggested to practice more and more questions. Aside from the sample paper you more likely had solved. These Determinants Class 12 MCQ are ready by the subject specialists themselves.
Question 11.The value of
is
(a) (a – b) (b – c) (c – a)
(b) (b – a) (c – b) (c – a)
(c) a (b – c) (c – a)
(d) None of these
A
Question 12. If x, y, z are all different from zero and
= 0, then value of x–1 + y–1 + z–1 is
(a) xyz
(b) x–1 y–1 z–1
(c) –x –y –z
(d) –1
D
Question 13.The value of the determinant
(a) 9×2(x + y)
(b) 9y2(x + y)
(c) 3y2(x + y)
(d) 7×2(x + y)
B
Question 14.The value of is
(a) 0
(b) 1
(c) –1
(d) None
A
Question 15.If area of triangle is 35 sq units with vertices (2, –6) (5, 4) and (k, 4), then k is
(a) 12
(b) –2
(c) –12, –2
(d) 12, –2
D
Question 16. If
+3 = 0 , then the value of x is
(a) 3
(b) 0
(c) –1
(d) 1
C
Question 17.If a, b, c are in AP, then the value of determinant Δ =
(a) 0
(b) 1
(c) x
(d) 2x
A
Question 18. If A is a non-singular square matrix of order 3 such that A2 = 3A, then value of |A| is
(a) –3
(b) 3
(c) 9
(d) 27
D
You can easily get good marks If you study with the help of Class 12 Determinants MCQ. We trust that information provided is useful for you. NCERT MCQ Questions for Class 12 Determinants PDF Free Download would without a doubt create positive results.
We hope the information shared above in regards to MCQ on Determinants Class 12 with Answers has been helpful to you. if you have any questions regarding CBSE Class 12 Mathematics Solutions MCQs Pdf, write a comment below and we will get back to you as soon as possible.<|endoftext|>
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# Water in a canal, 30 dm wide and 12 dm deep, is flowing with a velocity of 20 km per hour. How much area will it irrigate,
Question:
Water in a canal, 30 dm wide and 12 dm deep, is flowing with a velocity of 20 km per hour. How much area will it irrigate, if 9 cm of standing water is densired?
Solution:
Width of the canal = 30 dm = 3 m (1 m = 10 dm)
Depth of the canal = 12 dm = 1.2 m
Speed of the water flow = 20 km/h = 20000 m/h
∴ Volume of water flowing out of the canal in 1 h = 3 × 1.2 × 20000 = 72000 m3
Height of standing water on field = 9 cm = 0.09 m (1 m = 100 cm)
Assume that water flows out of the canal for 1 h. Then,
Area of the field irrigated
$=\frac{\text { Volume of water flowing out of the canal }}{\text { Height of standing water on the field }}$
$=\frac{72000}{0.09}$
$=800000 \mathrm{~m}^{2}$
$=\frac{800000}{10000} \quad\left(1\right.$ hectare $\left.=10000 \mathrm{~m}^{2}\right)$
$=80$ hectare
Thus, the area of the field irrigated is 80 hectares.
Disclaimer: In this question time is not given, so the question is solved assuming that the water flows out of the canal for 1 hour.<|endoftext|>
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To reflect a form over an axis, you have the right to either match the distance of a point to the axis top top the other side of utilizing the reflection notation.
You are watching: Reflection over the y axis rule
To enhance the distance, you can count the variety of units to the axis and also plot a allude on the corresponding allude over the axis.
You can likewise negate the value depending upon the line of reflection where the x-value is negate if the have fun is over the y-axis and the y-value is negated if the reflection is over the x-axis.
Either way, the answer is the very same thing.
For example:Triangle ABC v coordinate points A(1,2), B(3,5), and also C(7,1). Recognize the name: coordinates points that the image after a reflection end the x-axis.
Since the reflection used is walking to it is in over the x-axis, that way negating the y-value. As a result, clues of the photo are going to be:A"(1,-2), B"(3,-5), and C"(7,-1)
By counting the units, we know that point A is situated two devices above the x-axis. Count two units below the x-axis and there is suggest A’. Do the exact same for the various other points and the points are alsoA"(1,-2), B"(3,-5), and also C"(7,-1)
Reflection Notation:rx-axis = (x,-y)ry-axis = (-x,y)
## Video-Lesson Transcript
In this lesson, we’ll go over reflect on a name: coordinates system. This will certainly involve changing the coordinates.
For example, shot to reflect end the
-axis.
We have actually triangle
v coordinates
We’re going to reflect it end the
-axis. We’re going to upper and lower reversal it over.
So we’ll carry out what we normally do. Just one suggest at a time.
Now,
is over
units from the
-axis so we’ll move it listed below the
-axis through
units.
This will certainly be the
.
Let’s execute the same for
. It’s
units above the
-axis so we’re walk to go
units below the
-axis. Notification that it’s tho in line with
.
This is currently
.
Look at allude
at
. It’s
point over the
-axis for this reason we’ll go
point below the
-axis.
So,
.
And just affix the points. Climate we have the right to see our reflection end the
-axis.
When us reflect end the
-axis, something happens to the coordinates.
The initial coordinates
change. The
coordinate remains the same yet the
coordinate is the exact same number however now it’s negative.
In showing over the
-axis, we’ll write
Now, the same thing goes for mirroring over the
-axis.
We’re going come reflect triangle
end the
-axis.
Similar to mirroring over the
-axis, we’ll just do one suggest at a time.
is
unit from the
-axis therefore we’ll move
beyond the
-axis.
So,
.
Let’s look in ~
at
. That way it’s
units from the
-axis therefore we’ll move
collaborates on the various other side of the
-axis.
Now,
.
Finally,
is in ~
therefore we’ll go
points beyond the
-axis.
We’ll have actually
.
Now, us can attract a triangle the is a have fun of triangle
over the
-axis.
Let’s look at at just how these works with changed.
Originally us have collaborates
but
remained the same.
Let’s recap.
The ascendancy of reflecting over the
-axis is
And for showing over the
-axis is
If friend reflect it over the
-axis,
coordinate stays the very same the various other coordinate becomes negative.
See more: Ba(Oh)2 Lewis Structure For The Ionic Compound, Barium Hydroxide
And showing over the
-axis,
coordinate remains the same while the other coordinate i do not care negative.<|endoftext|>
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2.6 Books and the internet as sources
Finally, let's come back to the different types of modern sources as indicated in Figure 1. Many of these types are familiar to you in one way or another, so we can be brief. The course A219 uses set books that students registered with the Open University are required to purchase. Three of them are clearly modern scholarship: The Oxford Companion to Classical Civilization (OCCC), A Brief History of Ancient Greece (BHAG) and Rome in the Late Republic (Beard and Crawford). The other two, The Odyssey and Pliny's Letters, are both translations of ancient sources. You are not expected to read these books in order to complete this course, but their details have been included in the References section in case you wish to find out more about this subject.
The OCCC is an encyclopedia; it has succinct entries providing information about aspects of the ancient world, and while it rarely goes into a detailed discussion of sources, it does identify the sources and attempt to give you a reliable starting point. BHAG and Beard and Crawford are textbooks. They present a larger-scale overview of our understanding of the topics in question (Greece and late republican Rome respectively). You will shortly do a couple of activities to familiarise yourself with the particular issues involved in using these sources.
The most detailed form of modern source is the scholarly article, often in an academic journal, or book-length monographs. The difference between such specialist work and textbooks is on a sliding scale, rather than hard and fast. Perhaps the most important difference is the degree to which the work in question advances scholarship and constitutes a contribution to the sum of knowledge about the Classical world. There is little, if anything, in BHAG that is new scholarship; the aim of BHAG is to summarise existing thought and present an easily accessible account. Beard and Crawford, by contrast, are at least one step closer to scholarly monographs; part of what they do is put forward new suggestions in a way that you won't find in BHAG
Yet another, particularly problematic, modern source is the internet. Students registered on A219 use some internet applications that have been designed for them, but OpenLearn is unable to provide these in this short extract of the course. However, it is possible to use the internet to find information that isn't provided by us. It is worth, therefore, pointing out the most distinctive feature of the internet as a source: anyone can publish more or less anything they like very easily. This has the great advantage of making an enormous amount of material accessible to you in a way it wasn't to previous generations. But it has the great disadvantage that there is often no quality assurance (or at least this is the case at the time of writing, in 2005). If you buy a book – and certainly if you're encouraged to buy a book as part of a course – you can at least hope that the publisher will have done some work to ensure a certain degree of accuracy; you can assess the credentials of the author and look for a bibliography and footnotes or endnotes. This isn't so easy to check when you use the internet. Often it is impossible to identify the author, and supporting sources may not be apparent. So you should be even more critically thoughtful about the accuracy of information if it is taken from the internet. If you have the time, you may wish to take a look at Safari, a guide to the information world (including the internet) provided by the Open University Library.
This brings us to the end of our overview of modern sources, and we can now go on to become familiar with the OCCC and BHAG
Have a look at the OCCC entry for ‘tourism’ (p.729).
Click on the link below to open the OCCC definition for 'tourism'.
As you read it, note down the sources, ancient and modern, that are referred to, and try to organise them using our classifications above. (Remember: we divided ancient sources into archaeology, visual arts, literature, historiography, and documents.) You will find that many sources are abbreviated. This is normal practice in academic publications. Usefully, the OCCC has a complete list of abbreviations on pp.xvii–xxiv (which you may well find helpful when consulting other books, too). If you have access to the OCCC, for example, in your local library, you may like to look up abbreviations that you don't understand. There will then probably still be sources left which you don't know how to classify, since you have not read them (in fact, some of them are really quite obscure). If you are pressed for time, simply add a category headed ‘don't know’ to your list of classifications. If you have more time, you could look up the cross-referenced entries to elsewhere in the OCCC and check what sort of sources they are. In any case, we'd like you to look up at least one of the sources you don't know, to help you familiarise yourself with the OCCC and the quality of the information and discussion it contains.
Our list is as follows:
Art history / archaeology: the colossi of Memnon and other pharaonic monuments.
Literature: Isocrates, Trapeziticus 17.4; Heraclides Criticus,
On the Cities in Greece; Pausanias.
Historiography: Herodotus 1.30; Pausanias.
Documents: Tebtunis Papyri 1.33; Greek and Roman graffiti.
Pfister, E (1951): an edition.
Austin 83: a translation.
Bagnall and Derow 58: an edition of a manuscript.
As we said, there is no reason why you should have been able to categorise all these sources, but we hope that you will have succeeded with a few of them! Now we'd like to add a few comments, with the aim of giving you an idea of what sort of book the OCCC is and how you will be able to use it.
The references are frequent, detailed and precise. The OCCC is very compressed, packing a lot of material into a short space. This means that you will often find it impossible to understand all the detail at first. In most cases, when working with the OCCC, it will not be necessary for you to master all the detail, but this shouldn't deter you. For example, we'd never heard of Isocrates’ work Trapeziticus before reading the ‘tourism’ entry, and we'd be surprised if many of the course team had! However, using the OCCC, you can find out more and more detail the more you dig. If you looked up some of the sources you didn't know, you have already started this process.
Most of the sources are ancient. This is partly to do with the history of the OCCC, which is based on The Oxford Classical Dictionary. One of the key differences is that The Oxford Classical Dictionary gives bibliographies of modern sources at the end of each entry, while the OCCC doesn't. This observation provides a useful reminder of the privileged status of primary evidence. To support the description of ancient tourism, the author felt it necessary to cite numerous primary and no secondary sources, only editions and translations of primary sources.
We noticed that there are more word-based sources than those based on material culture. The only reference to material culture we found is in general terms to the ‘colossi of Memnon and other pharaonic monuments’. To explain this imbalance, we would need to go into more detail about the subject. It might be that more sources about tourism survive from literature, historiography and philosophy than from art and archaeology. Alternatively, the author may have a preference for written sources; after all, it is the written graffiti and literary associations that get mentioned in the caption more than the statues themselves. It may be the case that the dictionary style, with short articles and little space for illustrations, makes it harder to integrate material culture. Another possible explanation is that less research has been undertaken on the material evidence for ancient tourism. Whichever of these factors come into play, we hope you, too, saw this imbalance.
Finally, some sources are referred to very specifically (‘Hdt 1.30’), while others are quite vague (‘colossi of Memnon and other pharaonic monuments’; ‘Pausanias’). Why is that? The precise references are about very specific facts, so for example the paragraph ‘Hdt 1.30’ starts:
For this reason, then – and also no doubt for the pleasure of foreign travel – Solon left home and, after a visit to the court of Amasis in Egypt, went to Sardis to see Croesus.
Croesus entertained him hospitably in the palace, and three or four days after his arrival instructed some servants to take him on a tour of the royal treasuries and point out the richness and magnificence of everything.
(Herodotus 1.30; trans. de Sélincourt)
The translation indicates that the ancient text precisely reports the fact provided in the OCCC entry. Meanwhile, the more general references are about recurring facts: there are many Greek and Latin graffiti. The work of Pausanias almost entirely consists of descriptions of parts of Greece he visited and was informed about. Wherever possible, give a precise reference, but you don't necessarily need to give such precise references when you refer to a widespread and well-known phenomenon.
So much, then, for the OCCC. Textbooks like BHAG are rather different kinds of sources and require different skills in using them. The next activity is aimed at introducing these skills, at the same time as giving you some initial practice in working with ancient sources.
Block 2 of A219 starts with a section on Aeschylus' play Persians, which is set in the context of the Persian invasion of Greece at the beginning of the fifth century BCE. By way of a sneak preview, we will focus in this activity on one aspect of one battle in the course of this invasion. For our purposes the merest outline of the context is enough. Xerxes, the Persian king, invaded Greece with a massive army, on both sea and land. The Greeks let him take some regions without putting up much resistance, but then confronted him at a narrow pass between the mountains and the sea, called Thermopylae. In the event they lost the battle, but only after intense fighting. Most of them, including their leader, the Spartan Leonidas, died. BHAG has an account of this battle on pp. 134–5. Read this account now.
Click on the link below to open an account of the Battle of Thermophylae.
We want to focus only on the very last section in this short paragraph, starting with the decapitation of Leonidas (‘On Xerxes’ orders … ‘ up to and including the two-line epitaph). Modern scholarship, we said earlier on, are based on ancient sources, and that's why they are often called ‘secondary’. The most important ancient, or ‘primary’, source here is the historian Herodotus, whom we have already mentioned a few times. His account of the Battle of Thermopylae is lengthy, which is why we concentrate on the aftermath of the battle. Two sections of his narrative are relevant to our passage: Book 7, Chapters 228 and 238, or 7.228 and 7.238 for short.
Next read these two sections of Herodotus. They are attached below in PDF format. There will be some detail (in particular names) that you don't know. This happens quite frequently when you read ancient texts, and isn't just because you may not have studied the Classical world before. We, too, regularly read ancient texts of which we find certain details hard to understand. The important thing is to try to use, understand and evaluate the texts nonetheless, to the degree that you are able.
Click on the link below to open passages from Herodotus.
Once you have read the two Herodotus passages, compare them to the passage at the end of the paragraph in BHAG, and reflect on the following questions:
What do you learn about the way you can use ancient sources like Herodotus?
What do you learn about the way you can use modern scholarship like BHAG?
No doubt your thoughts aren't quite the same as ours. This is unavoidable in activities that demand a good deal of personal judgement (and, as we said before, that's the case for the majority of activities in this course). But we hope that there are enough points of contact between your way of approaching the question and ours to make our discussion useful to you. We will take the questions one by one.
1. One obvious (perhaps all too obvious) point to make is that the two Herodotus passages treating the events in question are separate. In writing their paragraph, the authors of BHAG had to collect these two different passages and put them together. This is one of the most fundamental aspects of using ancient sources: you will need to collect different sources and put them together. In this case, the two different sources come from the same overall source (Herodotus). In other cases, they will come from different places altogether.
Collecting, however, is only one aspect of what the BHAG authors did here. They also selected. Both passages had a lot of detail that didn't make it into BHAG: three little poems of which BHAG only prints the second one, for instance, and Herodotus' views on how Persians usually treat their enemies after battle, which weren't selected for inclusion in the textbook. The BHAG authors will have studied both passages, decided which detail is particularly important or relevant for their overall account of the battle at Thermopylae, and made their selection accordingly. This close study followed by selection is a standard practice in using ancient sources.
Related to this selection process is the issue of evaluation. Evaluation isn't explicit in the BHAG passage but can still be felt rather faintly. For instance, the BHAG authors must have decided that some basic details in Herodotus were correct, such as the epitaph they quote and Xerxes' orders to decapitate Leonidas. Perhaps (but this is pure speculation) they decided that other aspects were less reliable and that's why they left them out (such as Herodotus' thoughts about the exceptionality of the treatment administered to Leonidas). And their rather cagey phrase ‘attributed to Simonides’ suggests that they don't have complete trust in the reliability of whatever source stated that the poet Simonides is the author of that epitaph. (That source is probably not Herodotus: he is rather vague about this issue, it seems. What do you think?) Evaluation, too, is an essential aspect of work with ancient sources. Wherever you use ancient sources, you have to ask yourself how reliable you think they are. The answer is of course different from case to case.
2. What does all this mean for the way you can use BHAG? Well, perhaps the most important thing is that you should have worked out by now that even a textbook like BHAG doesn't simply tell you ‘the facts’ but makes its own choices about what to collect, select, suppress, trust, distrust and so on. The choices may be good (as they usually are) or they may be less good (as they sometimes are – no book is perfect), but they are always choices. Going back to the ancient sources will always tell you further things, and is therefore crucial whenever you want to get to the bottom of something.
Next, you have become familiar with some of the habits of BHAG. In particular, the authors are rather silent about how they collect, select and evaluate. The reason you now know which Herodotus passages are at the bottom of their account is that we worked it out for you (you could have done it yourself, but either way the point is that BHAG doesn't tell you). Other books are different, and you, as a matter of course, should be much more explicit than BHAG. Another aspect in which BHAG is silent is the rationale for its choices. Why the second of the three epigrams? Perhaps because it is the most famous one? (There are many later imitations, both in antiquity and in modern literature.) Or because they think it captures something about the Battle of Thermopylae that the others don't? Or do they like the uplifting tone of it and want us to go away with a rather heroic version of the Greeks at Thermopylae? And why do they leave out Herodotus’ comment that the Persians usually treat their enemies with more respect? Don't they believe it? Or do they want to have only information specific to Thermopylae here? Or do they want to create a crueller image of Xerxes? Again, we don't know. Modern scholars, just like ancient sources, have a bias, and it is important to think about bias when dealing with both kinds of source.
As we have pointed to some issues to be aware of when reading BHAG, we should probably stress that none of this is intended to warn you against using BHAG. On the contrary, the last two activities were designed to get you into the habit of using both the OCCC and BHAG as much as possible, but to use them critically. A219 students are asked throughout the course to use the OCCC, and throughout the Greek part to use BHAG. In their different ways, they are a mine of information – the OCCC for condensed accounts of numerous aspects of the Classical world, and BHAG for a more expansive narrative of Greek history. As should have become clear, they can't be treated as the last word, but they can be a good summary and a pointer to where to find out more.<|endoftext|>
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This article needs additional citations for verification. (November 2009) (Learn how and when to remove this template message)
Fiber-optic communication is a method of transmitting information from one place to another by sending pulses of light through an optical fiber. The light forms an electromagnetic carrier wave that is modulated to carry information. Fiber is preferred over electrical cabling when high bandwidth, long distance, or immunity to electromagnetic interference are required.
Optical fiber is used by many telecommunications companies to transmit telephone signals, Internet communication, and cable television signals. Researchers at Bell Labs have reached internet speeds of over 100 petabit×kilometer per second using fiber-optic communication.
- 1 Background
- 2 Applications
- 3 History
- 4 Technology
- 5 Parameters
- 6 Comparison with electrical transmission
- 7 Governing standards
- 8 See also
- 9 References
- 10 Further reading
- 11 External links
First developed in the 1970s, fiber-optics have revolutionized the telecommunications industry and have played a major role in the advent of the Information Age. Because of its advantages over electrical transmission, optical fibers have largely replaced copper wire communications in core networks in the developed world.
The process of communicating using fiber-optics involves the following basic steps:
- creating the optical signal involving the use of a transmitter, usually from an electrical signal
- relaying the signal along the fiber, ensuring that the signal does not become too distorted or weak
- receiving the optical signal
- converting it into an electrical signal
Optical fiber is used by many telecommunications companies to transmit telephone signals, Internet communication and cable television signals. Due to much lower attenuation and interference, optical fiber has large advantages over existing copper wire in long-distance, high-demand applications. However, infrastructure development within cities was relatively difficult and time-consuming, and fiber-optic systems were complex and expensive to install and operate. Due to these difficulties, fiber-optic communication systems have primarily been installed in long-distance applications, where they can be used to their full transmission capacity, offsetting the increased cost. The prices of fiber-optic communications have dropped considerably since 2000.
The price for rolling out fiber to homes has currently become more cost-effective than that of rolling out a copper based network. Prices have dropped to $850 per subscriber in the US and lower in countries like The Netherlands, where digging costs are low and housing density is high.
Since 1990, when optical-amplification systems became commercially available, the telecommunications industry has laid a vast network of intercity and transoceanic fiber communication lines. By 2002, an intercontinental network of 250,000 km of submarine communications cable with a capacity of 2.56 Tb/s was completed, and although specific network capacities are privileged information, telecommunications investment reports indicate that network capacity has increased dramatically since 2004.
In 1880 Alexander Graham Bell and his assistant Charles Sumner Tainter created a very early precursor to fiber-optic communications, the Photophone, at Bell's newly established Volta Laboratory in Washington, D.C. Bell considered it his most important invention. The device allowed for the transmission of sound on a beam of light. On June 3, 1880, Bell conducted the world's first wireless telephone transmission between two buildings, some 213 meters apart. Due to its use of an atmospheric transmission medium, the Photophone would not prove practical until advances in laser and optical fiber technologies permitted the secure transport of light. The Photophone's first practical use came in military communication systems many decades later.
In 1954 Harold Hopkins and Narinder Singh Kapany showed that rolled fiber glass allowed light to be transmitted. Initially it was considered that the light can traverse in only straight medium.[clarification needed]
Jun-ichi Nishizawa, a Japanese scientist at Tohoku University, proposed the use of optical fibers for communications in 1963. Nishizawa invented the PIN diode and the static induction transistor, both of which contributed to the development of optical fiber communications.
In 1966 Charles K. Kao and George Hockham at STC Laboratories (STL) showed that the losses of 1,000 dB/km in existing glass (compared to 5–10 dB/km in coaxial cable) were due to contaminants which could potentially be removed.
Optical fiber was successfully developed in 1970 by Corning Glass Works, with attenuation low enough for communication purposes (about 20 dB/km) and at the same time GaAs semiconductor lasers were developed that were compact and therefore suitable for transmitting light through fiber optic cables for long distances.
After a period of research starting from 1975, the first commercial fiber-optic communications system was developed which operated at a wavelength around 0.8 µm and used GaAs semiconductor lasers. This first-generation system operated at a bit rate of 45 Mbit/s with repeater spacing of up to 10 km. Soon on 22 April 1977, General Telephone and Electronics sent the first live telephone traffic through fiber optics at a 6 Mbit/s throughput in Long Beach, California.
In October 1973, Corning Glass signed a development contract with CSELT and Pirelli aimed to test fiber optics in an urban environment: in September 1977, the second cable in this test series, named COS-2, was experimentally deployed in two lines (9 km) in Turin, for the first time in a big city, at a speed of 140 Mbit/s.
The second generation of fiber-optic communication was developed for commercial use in the early 1980s, operated at 1.3 µm and used InGaAsP semiconductor lasers. These early systems were initially limited by multi mode fiber dispersion, and in 1981 the single-mode fiber was revealed to greatly improve system performance, however practical connectors capable of working with single mode fiber proved difficult to develop. Canadian service provider SaskTel had completed construction of what was then the world’s longest commercial fiber optic network, which covered 3,268 km and linked 52 communities. By 1987, these systems were operating at bit rates of up to 1.7 Gb/s with repeater spacing up to 50 km.
Third-generation fiber-optic systems operated at 1.55 µm and had losses of about 0.2 dB/km. This development was spurred by the discovery of Indium gallium arsenide and the development of the Indium Gallium Arsenide photodiode by Pearsall. Engineers overcame earlier difficulties with pulse-spreading at that wavelength using conventional InGaAsP semiconductor lasers. Scientists overcame this difficulty by using dispersion-shifted fibers designed to have minimal dispersion at 1.55 µm or by limiting the laser spectrum to a single longitudinal mode. These developments eventually allowed third-generation systems to operate commercially at 2.5 Gbit/s with repeater spacing in excess of 100 km.
The fourth generation of fiber-optic communication systems used optical amplification to reduce the need for repeaters and wavelength-division multiplexing to increase data capacity. These two improvements caused a revolution that resulted in the doubling of system capacity every six months starting in 1992 until a bit rate of 10 Tb/s was reached by 2001. In 2006 a bit-rate of 14 Tbit/s was reached over a single 160 km line using optical amplifiers.
The focus of development for the fifth generation of fiber-optic communications is on extending the wavelength range over which a WDM system can operate. The conventional wavelength window, known as the C band, covers the wavelength range 1.53–1.57 µm, and dry fiber has a low-loss window promising an extension of that range to 1.30–1.65 µm. Other developments include the concept of "optical solitons", pulses that preserve their shape by counteracting the effects of dispersion with the nonlinear effects of the fiber by using pulses of a specific shape.
In the late 1990s through 2000, industry promoters, and research companies such as KMI, and RHK predicted massive increases in demand for communications bandwidth due to increased use of the Internet, and commercialization of various bandwidth-intensive consumer services, such as video on demand. Internet protocol data traffic was increasing exponentially, at a faster rate than integrated circuit complexity had increased under Moore's Law. From the bust of the dot-com bubble through 2006, however, the main trend in the industry has been consolidation of firms and offshoring of manufacturing to reduce costs. Companies such as Verizon and AT&T have taken advantage of fiber-optic communications to deliver a variety of high-throughput data and broadband services to consumers' homes.
Modern fiber-optic communication systems generally include an optical transmitter to convert an electrical signal into an optical signal to send through the optical fiber, a cable containing bundles of multiple optical fibers that is routed through underground conduits and buildings, multiple kinds of amplifiers, and an optical receiver to recover the signal as an electrical signal. The information transmitted is typically digital information generated by computers, telephone systems and cable television companies.
The most commonly used optical transmitters are semiconductor devices such as light-emitting diodes (LEDs) and laser diodes. The difference between LEDs and laser diodes is that LEDs produce incoherent light, while laser diodes produce coherent light. For use in optical communications, semiconductor optical transmitters must be designed to be compact, efficient and reliable, while operating in an optimal wavelength range and directly modulated at high frequencies.
In its simplest form, an LED is a forward-biased p-n junction, emitting light through spontaneous emission, a phenomenon referred to as electroluminescence. The emitted light is incoherent with a relatively wide spectral width of 30–60 nm. LED light transmission is also inefficient, with only about 1% of input power, or about 100 microwatts, eventually converted into launched power which has been coupled into the optical fiber. However, due to their relatively simple design, LEDs are very useful for low-cost applications.
Communications LEDs are most commonly made from Indium gallium arsenide phosphide (InGaAsP) or gallium arsenide (GaAs). Because InGaAsP LEDs operate at a longer wavelength than GaAs LEDs (1.3 micrometers vs. 0.81–0.87 micrometers), their output spectrum, while equivalent in energy is wider in wavelength terms by a factor of about 1.7. The large spectrum width of LEDs is subject to higher fiber dispersion, considerably limiting their bit rate-distance product (a common measure of usefulness). LEDs are suitable primarily for local-area-network applications with bit rates of 10–100 Mbit/s and transmission distances of a few kilometers. LEDs have also been developed that use several quantum wells to emit light at different wavelengths over a broad spectrum and are currently in use for local-area WDM (Wavelength-Division Multiplexing) networks.
Today, LEDs have been largely superseded by VCSEL (Vertical Cavity Surface Emitting Laser) devices, which offer improved speed, power and spectral properties, at a similar cost. Common VCSEL devices couple well to multi mode fiber.
A semiconductor laser emits light through stimulated emission rather than spontaneous emission, which results in high output power (~100 mW) as well as other benefits related to the nature of coherent light. The output of a laser is relatively directional, allowing high coupling efficiency (~50 %) into single-mode fiber. The narrow spectral width also allows for high bit rates since it reduces the effect of chromatic dispersion. Furthermore, semiconductor lasers can be modulated directly at high frequencies because of short recombination time.
Laser diodes are often directly modulated, that is the light output is controlled by a current applied directly to the device. For very high data rates or very long distance links, a laser source may be operated continuous wave, and the light modulated by an external device, an optical modulator, such as an electro-absorption modulator or Mach–Zehnder interferometer. External modulation increases the achievable link distance by eliminating laser chirp, which broadens the linewidth of directly modulated lasers, increasing the chromatic dispersion in the fiber. For very high bandwidth efficiency, coherent modulation can be used to vary the phase of the light in addition to the amplitude, enabling the use of QPSK, QAM, and OFDM.
A transceiver is a device combining a transmitter and a receiver in a single housing (see picture on right).
Fiber optics have seen recent advances in technology. "Dual-polarization quadrature phase shift keying is a modulation format that effectively sends four times as much information as traditional optical transmissions of the same speed."
The main component of an optical receiver is a photodetector which converts light into electricity using the photoelectric effect. The primary photodetectors for telecommunications are made from Indium gallium arsenide. The photodetector is typically a semiconductor-based photodiode. Several types of photodiodes include p-n photodiodes, p-i-n photodiodes, and avalanche photodiodes. Metal-semiconductor-metal (MSM) photodetectors are also used due to their suitability for circuit integration in regenerators and wavelength-division multiplexers.
Optical-electrical converters are typically coupled with a transimpedance amplifier and a limiting amplifier to produce a digital signal in the electrical domain from the incoming optical signal, which may be attenuated and distorted while passing through the channel. Further signal processing such as clock recovery from data (CDR) performed by a phase-locked loop may also be applied before the data is passed on.
Coherent receivers use a local oscillator laser in combination with a pair of hybrid couplers and four photodetectors per polarization, followed by high speed ADCs and digital signal processing to recover data modulated with QPSK, QAM, or OFDM.
An optical communication system transmitter consists of a digital-to-analog converter (DAC), a driver amplifier and a Mach–Zehnder-Modulator. The deployment of higher modulation formats (> 4QAM) or higher Baud rates (> 32 GBaud) diminishes the system performance due to linear and non-linear transmitter effects. These effects can be categorised in linear distortions due to DAC bandwidth limitation and transmitter I/Q skew as well as non-linear effects caused by gain saturation in the driver amplifier and the Mach–Zehnder modulator. Digital predistortion counteracts the degrading effects and enables Baud rates up to 56 GBaud and modulation formats like 64QAM and 128QAM with the commercially available components. The transmitter digital signal processor performs digital predistortion on the input signals using the inverse transmitter model before uploading the samples to the DAC.
Older digital predistortion methods only addressed linear effects. Recent publications also compensated for non-linear distortions. Berenguer et al models the Mach–Zehnder modulator as an independent Wiener system and the DAC and the driver amplifier are modelled by a truncated, time-invariant Volterra series. Khanna et al used a memory polynomial to model the transmitter components jointly. In both approaches the Volterra series or the memory polynomial coefficients are found using Indirect-learning architecture. Duthel et al records for each branch of the Mach-Zehnder modulator several signals at different polarity and phases. The signals are used to calculate the optical field. Cross-correlating in-phase and quadrature fields identifies the timing skew. The frequency response and the non-linear effects are determined by the indirect-learning architecture.
Fiber cable types
An optical fiber cable consists of a core, cladding, and a buffer (a protective outer coating), in which the cladding guides the light along the core by using the method of total internal reflection. The core and the cladding (which has a lower-refractive-index) are usually made of high-quality silica glass, although they can both be made of plastic as well. Connecting two optical fibers is done by fusion splicing or mechanical splicing and requires special skills and interconnection technology due to the microscopic precision required to align the fiber cores.
Two main types of optical fiber used in optic communications include multi-mode optical fibers and single-mode optical fibers. A multi-mode optical fiber has a larger core (≥ 50 micrometers), allowing less precise, cheaper transmitters and receivers to connect to it as well as cheaper connectors. However, a multi-mode fiber introduces multimode distortion, which often limits the bandwidth and length of the link. Furthermore, because of its higher dopant content, multi-mode fibers are usually expensive and exhibit higher attenuation. The core of a single-mode fiber is smaller (<10 micrometers) and requires more expensive components and interconnection methods, but allows much longer, higher-performance links. Both single- and multi-mode fiber is offered in different grades.
@850 nm &
In order to package fiber into a commercially viable product, it typically is protectively coated by using ultraviolet (UV), light-cured acrylate polymers, then terminated with optical fiber connectors, and finally assembled into a cable. After that, it can be laid in the ground and then run through the walls of a building and deployed aerially in a manner similar to copper cables. These fibers require less maintenance than common twisted pair wires once they are deployed.
Specialized cables are used for long distance subsea data transmission, e.g. transatlantic communications cable. New (2011–2013) cables operated by commercial enterprises (Emerald Atlantis, Hibernia Atlantic) typically have four strands of fiber and cross the Atlantic (NYC-London) in 60–70ms. Cost of each such cable was about $300M in 2011. source: The Chronicle Herald.
Another common practice is to bundle many fiber optic strands within long-distance power transmission cable. This exploits power transmission rights of way effectively, ensures a power company can own and control the fiber required to monitor its own devices and lines, is effectively immune to tampering, and simplifies the deployment of smart grid technology.
The transmission distance of a fiber-optic communication system has traditionally been limited by fiber attenuation and by fiber distortion. By using opto-electronic repeaters, these problems have been eliminated. These repeaters convert the signal into an electrical signal, and then use a transmitter to send the signal again at a higher intensity than was received, thus counteracting the loss incurred in the previous segment. Because of the high complexity with modern wavelength-division multiplexed signals (including the fact that they had to be installed about once every 20 km), the cost of these repeaters is very high.
An alternative approach is to use optical amplifiers which amplify the optical signal directly without having to convert the signal to the electrical domain. One common type of optical amplifier is called an Erbium-doped fiber amplifier, or EDFA. These are made by doping a length of fiber with the rare-earth mineral erbium and pumping it with light from a laser with a shorter wavelength than the communications signal (typically 980 nm). EDFAs provide gain in the ITU C band at 1550 nm, which is near the loss minimum for optical fiber.
Optical amplifiers have several significant advantages over electrical repeaters. First, an optical amplifier can amplify a very wide band at once which can include hundreds of individual channels, eliminating the need to demultiplex DWDM signals at each amplifier. Second, optical amplifiers operate independently of the data rate and modulation format, enabling multiple data rates and modulation formats to co-exist and enabling upgrading of the data rate of a system without having to replace all of the repeaters. Third, optical amplifiers are much simpler than a repeater with the same capabilities and are therefore significantly more reliable. Optical amplifiers have largely replaced repeaters in new installations, although electronic repeaters are still widely used as transponders for wavelength conversion.
Wavelength-division multiplexing (WDM) is the practice of multiplying the available capacity of optical fibers through use of parallel channels, each channel on a dedicated wavelength of light. This requires a wavelength division multiplexer in the transmitting equipment and a demultiplexer (essentially a spectrometer) in the receiving equipment. Arrayed waveguide gratings are commonly used for multiplexing and demultiplexing in WDM. Using WDM technology now commercially available, the bandwidth of a fiber can be divided into as many as 160 channels to support a combined bit rate in the range of 1.6 Tbit/s.
Because the effect of dispersion increases with the length of the fiber, a fiber transmission system is often characterized by its bandwidth–distance product, usually expressed in units of MHz·km. This value is a product of bandwidth and distance because there is a trade-off between the bandwidth of the signal and the distance over which it can be carried. For example, a common multi-mode fiber with bandwidth–distance product of 500 MHz·km could carry a 500 MHz signal for 1 km or a 1000 MHz signal for 0.5 km.
Engineers are always looking at current limitations in order to improve fiber-optic communication, and several of these restrictions are currently being researched.
Each fiber can carry many independent channels, each using a different wavelength of light (wavelength-division multiplexing). The net data rate (data rate without overhead bytes) per fiber is the per-channel data rate reduced by the FEC overhead, multiplied by the number of channels (usually up to eighty in commercial dense WDM systems as of 2008[update]).
Standard fibre cables
The following summarizes the current state-of-the-art research using standard telecoms-grade single-mode, single-solid-core fibre cables.
|Year||Organization||Effective speed||WDM channels||Per channel speed||Distance|
|2009||Alcatel-Lucent||15.5 Tbit/s||155||100 Gbit/s||7000 km|
|2010||NTT||69.1 Tbit/s||432||171 Gbit/s||240 km|
|2011||NEC||101.7 Tbit/s||370||273 Gbit/s||165 km|
|2011||KIT||26 Tbit/s||>300||50 km|
|2016||BT & Huawei||5.6 Tbit/s
||28||200Gb/s||circa 140 km ?|
|2016||Nokia Bell Labs, Deutsche Telekom T-Labs & Technical University of Munich||1 Tbit/s
|2017||BT & Huawei||11.2 Tbit/s
||28||400 Gb/s||250 Km|
The 2016 Nokia/DT/TUM result is notable as it is the first result that pushes close to the Shannon theoretical limit.
The following summaries the current state-of-the-art research using specialised cables that allow spatial multiplexing to occur, use specialised tri-mode fibre cables or similar specialised fibre optic cables.
|Year||Organization||Effective speed||No. of propagation modes||No. of cores||WDM channels (per core)||Per channel speed||Distance|
|2012||NEC, Corning||1.05 Pbit/s||12||52.4 km|
|2013||University of Southampton||73.7 Tbit/s||1 (hollow)||3x96
|256 Gb/s||310 m|
|2014||Technical University of Denmark||43 Tbit/s||7||1045 km|
|2014||Eindhoven University of Technology (TU/e) and University of Central Florida (CREOL)||255 Tbit/s||7||50||~728 Gb/s||1 km|
|2015||NICT, Sumitomo Electric and RAM Photonics||2.15 Pbit/s||22||402 (C+L bands)||243 Gb/s||31 km|
|2017||NTT||1 Pbit/s||single-mode||32||46||680 Gb/s||205.6 Km|
|2017||KDDI Research and Sumitomo Electric||10.16 Pbit/s||6-mode||19||739 (C+L bands)||120 Gb/s||11.3 Km|
|2018||NICT||159 Tbit/s||tri-mode||1||348||414 Gb/s||1045 km|
The 2018 NICT result is notable for breaking the record for throughput using a single core cable, that is, not using spatial multiplexing.
Research from DTU, Fujikura & NTT is notable in that the team were able to reduce the power consumption of the optics to around 5% compared with more mainstream techniques, which could lead to a new generation of very power efficient optic components.
|Year||Organization||Effective speed||No. of cores||WDM channels (per core)||Per channel speed||Distance|
|2018||Hao Hu, et al (DTU, Fujikura & NTT)||768 Tbit/s
Research conducted by the RMIT University, Melbourne, Australia, have developed a nanophotonic device that has achieved a 100 fold increase in current attainable fiber optic speeds by using a twisted-light technique. This technique carries data on light waves that have been twisted into a spiral form, to increase the optic cable capacity further, this technique is known as orbital angular momentum (OAM). The nanophotonic device uses ultra thin topological nanosheets to measure a fraction of a millimeter of twisted light, the nano-electronic device is embedded within a connector smaller then the size of a USB connector, it fits easily at the end of a optical fiber cable. The device can also be used to receive quantum information sent via twisted light, it is likely to be used in a new range of quantum communication and quantum computing research.
For modern glass optical fiber, the maximum transmission distance is limited not by direct material absorption but by several types of dispersion, or spreading of optical pulses as they travel along the fiber. Dispersion in optical fibers is caused by a variety of factors. Intermodal dispersion, caused by the different axial speeds of different transverse modes, limits the performance of multi-mode fiber. Because single-mode fiber supports only one transverse mode, intermodal dispersion is eliminated.
In single-mode fiber performance is primarily limited by chromatic dispersion (also called group velocity dispersion), which occurs because the index of the glass varies slightly depending on the wavelength of the light, and light from real optical transmitters necessarily has nonzero spectral width (due to modulation). Polarization mode dispersion, another source of limitation, occurs because although the single-mode fiber can sustain only one transverse mode, it can carry this mode with two different polarizations, and slight imperfections or distortions in a fiber can alter the propagation velocities for the two polarizations. This phenomenon is called fiber birefringence and can be counteracted by polarization-maintaining optical fiber. Dispersion limits the bandwidth of the fiber because the spreading optical pulse limits the rate that pulses can follow one another on the fiber and still be distinguishable at the receiver.
Some dispersion, notably chromatic dispersion, can be removed by a 'dispersion compensator'. This works by using a specially prepared length of fiber that has the opposite dispersion to that induced by the transmission fiber, and this sharpens the pulse so that it can be correctly decoded by the electronics.
Fiber attenuation, which necessitates the use of amplification systems, is caused by a combination of material absorption, Rayleigh scattering, Mie scattering, and connection losses. Although material absorption for pure silica is only around 0.03 dB/km (modern fiber has attenuation around 0.3 dB/km), impurities in the original optical fibers caused attenuation of about 1000 dB/km. Other forms of attenuation are caused by physical stresses to the fiber, microscopic fluctuations in density, and imperfect splicing techniques.
Each effect that contributes to attenuation and dispersion depends on the optical wavelength. There are wavelength bands (or windows) where these effects are weakest, and these are the most favorable for transmission. These windows have been standardized, and the currently defined bands are the following:
|O band||original||1260 to 1360 nm|
|E band||extended||1360 to 1460 nm|
|S band||short wavelengths||1460 to 1530 nm|
|C band||conventional ("erbium window")||1530 to 1565 nm|
|L band||long wavelengths||1565 to 1625 nm|
|U band||ultralong wavelengths||1625 to 1675 nm|
Note that this table shows that current technology has managed to bridge the second and third windows that were originally disjoint.
Historically, there was a window used below the O band, called the first window, at 800–900 nm; however, losses are high in this region so this window is used primarily for short-distance communications. The current lower windows (O and E) around 1300 nm have much lower losses. This region has zero dispersion. The middle windows (S and C) around 1500 nm are the most widely used. This region has the lowest attenuation losses and achieves the longest range. It does have some dispersion, so dispersion compensator devices are used to remove this.
When a communications link must span a larger distance than existing fiber-optic technology is capable of, the signal must be regenerated at intermediate points in the link by optical communications repeaters. Repeaters add substantial cost to a communication system, and so system designers attempt to minimize their use.
Recent advances in fiber and optical communications technology have reduced signal degradation so far that regeneration of the optical signal is only needed over distances of hundreds of kilometers. This has greatly reduced the cost of optical networking, particularly over undersea spans where the cost and reliability of repeaters is one of the key factors determining the performance of the whole cable system. The main advances contributing to these performance improvements are dispersion management, which seeks to balance the effects of dispersion against non-linearity; and solitons, which use nonlinear effects in the fiber to enable dispersion-free propagation over long distances.
Although fiber-optic systems excel in high-bandwidth applications, optical fiber has been slow to achieve its goal of fiber to the premises or to solve the last mile problem. However, as bandwidth demand increases, more and more progress towards this goal can be observed. In Japan, for instance EPON has largely replaced DSL as a broadband Internet source. South Korea’s KT also provides a service called FTTH (Fiber To The Home), which provides fiber-optic connections to the subscriber’s home. The largest FTTH deployments are in Japan, South Korea, and China. Singapore started implementation of their all-fiber Next Generation Nationwide Broadband Network (Next Gen NBN), which is slated for completion in 2012 and is being installed by OpenNet. Since they began rolling out services in September 2010, network coverage in Singapore has reached 85% nationwide.
In the US, Verizon Communications provides a FTTH service called FiOS to select high-ARPU (Average Revenue Per User) markets within its existing territory. The other major surviving ILEC (or Incumbent Local Exchange Carrier), AT&T, uses a FTTN (Fiber To The Node) service called U-verse with twisted-pair to the home. Their MSO competitors employ FTTN with coax using HFC. All of the major access networks use fiber for the bulk of the distance from the service provider's network to the customer.
The globally dominant access network technology is EPON (Ethernet Passive Optical Network). In Europe, and among telcos in the United States, BPON (ATM-based Broadband PON) and GPON (Gigabit PON) had roots in the FSAN (Full Service Access Network) and ITU-T standards organizations under their control.
Comparison with electrical transmission
The choice between optical fiber and electrical (or copper) transmission for a particular system is made based on a number of trade-offs. Optical fiber is generally chosen for systems requiring higher bandwidth or spanning longer distances than electrical cabling can accommodate.
The main benefits of fiber are its exceptionally low loss (allowing long distances between amplifiers/repeaters), its absence of ground currents and other parasite signal and power issues common to long parallel electric conductor runs (due to its reliance on light rather than electricity for transmission, and the dielectric nature of fiber optic), and its inherently high data-carrying capacity. Thousands of electrical links would be required to replace a single high bandwidth fiber cable. Another benefit of fibers is that even when run alongside each other for long distances, fiber cables experience effectively no crosstalk, in contrast to some types of electrical transmission lines. Fiber can be installed in areas with high electromagnetic interference (EMI), such as alongside utility lines, power lines, and railroad tracks. Nonmetallic all-dielectric cables are also ideal for areas of high lightning-strike incidence.
For comparison, while single-line, voice-grade copper systems longer than a couple of kilometers require in-line signal repeaters for satisfactory performance; it is not unusual for optical systems to go over 100 kilometers (62 mi), with no active or passive processing. Single-mode fiber cables are commonly available in 12 km lengths, minimizing the number of splices required over a long cable run. Multi-mode fiber is available in lengths up to 4 km, although industrial standards only mandate 2 km unbroken runs.
In short distance and relatively low bandwidth applications, electrical transmission is often preferred because of its
- Lower material cost, where large quantities are not required
- Lower cost of transmitters and receivers
- Capability to carry electrical power as well as signals (in appropriately designed cables)
- Ease of operating transducers in linear mode.
Optical fibers are more difficult and expensive to splice than electrical conductors. And at higher powers, optical fibers are susceptible to fiber fuse, resulting in catastrophic destruction of the fiber core and damage to transmission components.
Because of these benefits of electrical transmission, optical communication is not common in short box-to-box, backplane, or chip-to-chip applications; however, optical systems on those scales have been demonstrated in the laboratory.
In certain situations fiber may be used even for short distance or low bandwidth applications, due to other important features:
- Immunity to electromagnetic interference, including nuclear electromagnetic pulses.
- High electrical resistance, making it safe to use near high-voltage equipment or between areas with different earth potentials.
- Lighter weight—important, for example, in aircraft.
- No sparks—important in flammable or explosive gas environments.
- Not electromagnetically radiating, and difficult to tap without disrupting the signal—important in high-security environments.
- Much smaller cable size—important where pathway is limited, such as networking an existing building, where smaller channels can be drilled and space can be saved in existing cable ducts and trays.
- Resistance to corrosion due to non-metallic transmission medium
Optical fiber cables can be installed in buildings with the same equipment that is used to install copper and coaxial cables, with some modifications due to the small size and limited pull tension and bend radius of optical cables. Optical cables can typically be installed in duct systems in spans of 6000 meters or more depending on the duct's condition, layout of the duct system, and installation technique. Longer cables can be coiled at an intermediate point and pulled farther into the duct system as necessary.
In order for various manufacturers to be able to develop components that function compatibly in fiber optic communication systems, a number of standards have been developed. The International Telecommunications Union publishes several standards related to the characteristics and performance of fibers themselves, including
- ITU-T G.651, "Characteristics of a 50/125 µm multimode graded index optical fibre cable"
- ITU-T G.652, "Characteristics of a single-mode optical fibre cable"
Other standards specify performance criteria for fiber, transmitters, and receivers to be used together in conforming systems. Some of these standards are:
- 100 Gigabit Ethernet
- 10 Gigabit Ethernet
- Fibre Channel
- Gigabit Ethernet
- Synchronous Digital Hierarchy
- Synchronous Optical Networking
- Optical Transport Network (OTN)
- Dark fiber
- Fiber to the x
- Free-space optical communication
- Information theory
- Submarine communications cable
- Passive optical network
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- "One Petabit per Second Fiber Transmission over a Record Distance of 200 km" (PDF). NTT. 2017-03-23. Retrieved 2018-06-30.
- "Success of ultra-high capacity optical fibre transmission breaking the world record by a factor of five and reaching 10 Petabits per second" (PDF). Global Sei. 2017-10-13. Retrieved 2018-08-25.
- "Researchers in Japan 'break transmission record' over 1,045km with three-mode optical fibre". fibre-systems.com. 2018-04-16. Retrieved 2018-06-30.
- "Single-source chip-based frequency comb enabling extreme parallel data transmission". Nature Photonics (volume 12, pages 469–473). 2018-07-02. Retrieved 2018-08-02.
- "Groundbreaking new technology could allow 100-times-faster internet by harnessing twisted light beams". Phys.org. 2018-10-24. Retrieved 2018-10-25.
- "Angular-momentum nanometrology in an ultrathin plasmonic topological insulator film". Nature Communications (volume 9, Article number: 4413). 2018-10-24. Retrieved 2018-10-25.
- Encyclopedia of Laser Physics and Technology
- Lee, M. M.; J. M. Roth; T. G. Ulmer; C. V. Cryan (2006). "The Fiber Fuse Phenomenon in Polarization-Maintaining Fibers at 1.55 μm" (PDF). Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies. paper JWB66. Optical Society of America. Archived from the original (PDF) on July 17, 2011. Retrieved March 14, 2010.
- McAulay, Alastair D. (2011). Military Laser Technology for Defense: Technology for Revolutionizing 21st Century Warfare. John Wiley & Sons. ISBN 9781118019542.
Optical sensors are advantageous in hazardous environments because there are no sparks when a fiber breaks or its cover is worn.
- Encyclopedia of Laser Physics and Technology
- Fiber-Optic Technologies by Vivek Alwayn
- Agrawal, Govind P. (2002). Fiber-optic communication systems. New York: John Wiley & Sons. ISBN 978-0-471-21571-4.
- Keiser, Gerd. (2011). Optical fiber communications, 4th ed. New York, NY: McGraw-Hill, ISBN 9780073380711
- Senior, John. (2008). Optical Fiber Communications: Principles and Practice, 3rd ed. Prentice Hall. ISBN 978-0130326812
This article's use of external links may not follow Wikipedia's policies or guidelines. (February 2013) (Learn how and when to remove this template message)
|Wikimedia Commons has media related to Fiber-optic communications.|
- How Fiber-optics work (Howstuffworks.com)
- The Laser and Fiber-optic Revolution
- Fiber Optics, from Hyperphysics at Georgia State University
- "Understanding Optical Communications" An IBM redbook
- FTTx Primer July 2008
- Fibre optic transmission in security and surveillance solutions
- Fiber Optics - Internet, Cable and Telephone Communication
- Simulation of fiber-based optical transmission systems<|endoftext|>
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During the period of the Stamp Act’s enforcement, colonial governments nullified the law and the people’s actions, as evidenced in the protests in Wilmington, cancelled the law. From 1765 to 1766, American colonists joined a group named Sons of Liberty. The association met similarly to colonial assemblies, yet each each chapter declared more revolutionary measures than the official assembly. In many ways, the Sons of Liberty’s existence reveals less dependence on the government and more on individual and community solutions to problems. A decade later, the Sons of Liberty and similar groups such as the Sons of Neptune and the Philadelphia Patriotic Society, according to historian Larry Schweikart, “provided the organizational framework necessary for revolution.”
After the Connecticut Sons of Liberty denounced the Stamp Act and pledged to fight, if necessary, the Sons of Liberty followed suit across the colonies. In Wilmington, North Carolina, Sons of Liberty members pledged to resist the tax “with [their] lives and fortunes.” As a means of protest, the Wilmington chapter held mock funeral processions. Sons of Liberty members undoubtedly ignited the 1766 protests against the closing of North Carolina ports, especially the Wilmington port.
The colonial protests and refutation of Sir William Blackstone’s argument of parliamentary sovereignty led to the repeal of the Stamp Act. The Sons of Liberty later played a major role persuading non-compliers to enforce a boycott of British goods.
Murray N. Rothbard, Conceived in Liberty Vol. III (Auburn, Alabama: reprint, 1999); Milton Ready, The Tar Heel State, A History of North Carolina (Columbia, South Carolina: 2005); Larry Schweikart and Michael Allen, A Patriot’s History of the United States (New York, New York: 2004).<|endoftext|>
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# Math Expressions Grade 5 Unit 8 Lesson 13 Answer Key Volume of Composite Solid Figures
## Math Expressions Common Core Grade 5 Unit 8 Lesson 13 Answer Key Volume of Composite Solid Figures
Math Expressions Grade 5 Unit 8 Lesson 13 Homework
Find the volume of each composite solid figure.
Volume Of Composite Figures Worksheet Math Expressions Grade 5 Pdf Question 1.
Volume Of Composite Figures Answer Key Math Expressions Grade 5 Unit 8 Question 2.
Composite Figures Volume Math Expressions Grade 5 Unit 8 Question 3.
The volume of the figure is 204 cu in.
Explanation:
Here, we will divide the figure into rectangles, and then we will find the volume of each rectangle, then we will add them. So the volume of the rectangle is
V = l × w × h,
V1 = 4 × 13 × 3
= 156 cu in.
V2 = 3 × 4 × 4
= 48 cu in.
So the volume of the solid figure is 156 + 48
= 204 cu in.
Volume Of Composite Figures Pdf Math Expressions Grade 5 Unit 8 Question 4.
The exterior of a refrigerator is shaped like a rectangular prism, and measures 2$$\frac{2}{3}$$ feet wide by 5$$\frac{1}{2}$$ feet high by 2$$\frac{1}{2}$$ feet deep. What amount of space does the refrigerator take up?
The amount of space does the refrigerator take-up is 36 $$\frac{2}{3}$$ cu ft.
Explanation:
Given that the exterior of a refrigerator is shaped like a rectangular prism, and measures 2$$\frac{2}{3}$$ feet wide by 5$$\frac{1}{2}$$ feet high by 2$$\frac{1}{2}$$ feet deep. So the amount of space does the refrigerator take-up is 2$$\frac{2}{3}$$ × 5$$\frac{1}{2}$$ × 2$$\frac{1}{2}$$
= $$\frac{8}{3}$$ × $$\frac{11}{2}$$ × $$\frac{5}{2}$$
= $$\frac{440}{12}$$
= 36 $$\frac{2}{3}$$ cu ft.
Volume Of Composite Math Expressions Grade 5 Unit 8 Question 5.
In the space below, draw a composite solid of your own design that is made up of two prisms. Write the dimensions of your design, and then calculate its volume.
Math Expressions Grade 5 Unit 8 Lesson 13 Remembering
Divide.
Volume Of Composite Figures Math Expressions Grade 5 Unit 8 Question 1.
= 70.
Explanation:
Here, we will change the divisor 0.7 to a whole number by moving the decimal point 1 place to the right, and then we will move the decimal point in the dividend to the same 1 place right.
Question 2.
= 1000.
Explanation:
Here, we will change the divisor 0.05 to a whole number by moving the decimal point 2 places to the right, and then we will move the decimal point in the dividend to the same 2 places right.
Question 3.
= 0.8
Explanation:
Here, we will change the divisor 0.8 to a whole number by moving the decimal point 1 place to the right, and then we will move the decimal point in the dividend to the same 1 place right.
Question 4.
= 600.
Explanation:
Here, we will change the divisor 0.06 to a whole number by moving the decimal point 2 places to the right, and then we will move the decimal point in the dividend to the same 2 places right.
Question 5.
= 3,132.
Explanation:
Here, we will change the divisor 0.3 to a whole number by moving the decimal point 1 place to the right, and then we will move the decimal point in the dividend to the same 1 place right.
Question 6.
= 455.
Explanation:
Here, we will change the divisor 0.06 to a whole number by moving the decimal point 2 places to the right, and then we will move the decimal point in the dividend to the same 2 places right.
Solve.
Question 7.
A fish tank is 20 feet long, 12 feet wide, and 10 feet deep. What is the volume of the fish tank?<|endoftext|>
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Shan (Tai Yai) period: 1315 – 1948
The Tai-Shan people are believed to have migrated from Yunnan in China.
The Shan, who call themselves Tai, form one section of the large Tai ethnic group which is now believed to have spread from Southeast China through Vietnam, Laos, Thailand, Burma and Assam. Shan legendsindicate that they were already in the part of Burma which they still inhabit as early as the mid 11th century. The word “Shan” comes from the same root as Siam; in Bagan they were known as the Syam.
Buddha statues from the Shan period
Although there is evidence of earlier occupation, the first Shan Buddha images that were found seem to date back only to the 17th century. They have triangular faces with a broad forehead, eyebrows arched high over narrowly opened eyes, a pointed nose with triangular nostrils, pursed thin lips, large and elongated ears, and short necks. They are often seated in Vajrasana with hands in Bhumisparsha mudra. Seventeenth century Buddha statues may be placed on high, wasted lotus thrones and wear immensely tall crowns with flamboyant ear flanges.
Discover more about
The image and its history
Art, Architecture and Design of Burma
Burmese Buddhist Sculpture
The Johan Möger Collection<|endoftext|>
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Luckily, according to a new study by physicists at Brown University, DNA molecules have a convenient tendency to cooperate.
The research, published in the journal Physical Review Letters, looks at the dynamics of how DNA molecules are captured by solid-state nanopores, tiny holes that soon may help sequence DNA at lightning speed. The study found that when a DNA strand is captured and pulled through a nanopore, it’s much more likely to start the journey at one of its ends, rather than being grabbed somewhere in the middle and pulled through in a folded configuration.
“We think this is an important advance for understanding how DNA molecules interact with these nanopores,” said Derek Stein, assistant professor of physics at Brown, who performed the research with graduate student Mirna Mihovilovic and undergraduate Nick Hagerty. “If you want to do sequencing or some other analysis, you want the molecule going through the pore head to tail.”
Research into DNA sequencing with nanopores started a little over 15 years ago. The concept is fairly simple. A little hole, a few billionths of a meter across, is poked in a barrier separating two pools of salt water. An electric current is applied across the hole, which occasionally attracts a DNA molecule floating in the water. When that happens, the molecule is whipped through the pore in a fraction of a second. Scientists can then use sensors on the pore or other means to identify nucleotide bases, the building blocks of the genetic code.
The technology is advancing quickly, and the first nanopore sequencing devices are expected to be on the market very soon. But there are still basic questions about how molecules behave at the moment they’re captured and before.
“What the molecules were doing before they’re captured was a mystery and a matter of speculation,” Stein said. “And we’d like to know because if you’re trying to engineer something to control that molecule — to get it to do what you want it to do — you need to know what it’s up to.”
To find out what those molecules are up to, the researchers carefully tracked over 1,000 instances of a molecule zipping through a nanopore. The electric current through the pore provides a signal of how the molecule went through. Molecules that go through middle first have to be folded over in order to pass. That folded configuration takes up more space in the pore and blocks more of the current. So by looking at differences in the current, Stein and his team could count how many molecules went through head first and how many started somewhere in the middle.
The study found that molecules are several times more likely to be captured at or very near an end than at any other single point along the molecule.
“What we found was that ends are special places,” Stein said. “The middle is different from an end, and that has a consequence for the likelihood a molecule starts its journey from the end or the middle.”
Always room for Jell-O
As it turns out, there’s an old theory that that explains these new experimental results quite well. It’s the theory of Jell-O.
Jell-O is a polymer network — a mass of squiggly polymer strands that attach to each other at random junctions. The squiggly strands are the reason Jell-O is a jiggly, semi-solid. The way in which the polymer strands connect to each other is not unlike the way a DNA strand connects to a nanopore in the instant it’s captured. In water, DNA molecules are jumbled up in random squiggles much like the gelatin molecules in Jell-O.
“There’s some powerful theory that describes how many ways the polymers in Jell-O can arrange and attach themselves,” Stein said. “That turns out to be perfectly applicable to the problem of where these DNA molecules get captured by a nanopore.”
When applied to DNA, the Jell-O theory predicts that if you were to count up all the possible configurations of a DNA strand at the moment of capture, you would find that there are more configurations in which it is captured by its end, compared to other points along the strand. It’s a bit like the odds of getting a pair in poker compared to the odds of getting three of a kind. You’re more likely to get a pair simply because there are more pairs in the deck than there are triples.
This measure of all the possible configurations — a measure of what physicists refer to as the molecule’s entropy — is all that’s needed to explain why DNA tends to go head first. Some scientists had speculated that perhaps strands would be less likely to go through by the middle because folding them in half would require extra energy. But that folding energy appears not to matter at all. As Stein puts it, “The number of ways that a molecule can find itself with its head sticking in the pore is simply larger than the number of ways it can find itself with the middle touching the pore.”
These theories of polymer networks have actually been around for a while. They were first proposed by the late Nobel laureate Pierre-Gilles de Gennes in the 1960s, and Bertrand Duplantier made key advances in the 1980s. Mihovilovic, Stein’s graduate student and the lead author of this study, says this is actually one of the first lab tests of those theories.
“They couldn’t be tested until now, when we can actually do single molecule measurements,” she said. “[De Gennes] postulated that one day it would be possible to test this. I think he would have been very excited to see it happen.”<|endoftext|>
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Every Question Helps You Learn
Are you good at maths? Find out in this quiz!
# Level 5-6 Algebra - Equations with Brackets
Algebra is a part of your KS3 Maths journey that you'll need to get the hang of. This quiz is all about equations (a big part of algebra), specifically ones with brackets. Get ready to figure out the value of x or other letters like a or b by shuffling around letters and numbers. It's like a beginner's guide to equations, helping you grasp the concept.
1.
3a + 6 + b = 3a + 5 is the same as which of the following?
b = 3a - 3a + 5 - 6
b = 3a + 3a + 5 - 6
b = 3a - 3a - 5 + 6
b = 3a - 3a + 5 + 6
The correct answer could be further simplified to b = -1
2.
3 x a + 4 = 6 is the same as which of the following?
3 x a = 6 - 4
3 x a = 6 / 4
3 x a = 6 + 4
3 x a = 6 x 4
When the 4 crosses the equals sign it changes from + 4 to - 4
3.
If 16(a + 7) = 128, what is the value of a?
1
2
3
4
16a + 112 = 128 and therefore 16a = 16
4.
5(a - 4) is the same as which of the following?
5a + 20
5a - 20
5a - 4
5a4
Removing brackets this way is referred to as 'expanding the brackets'. In this case the 5 outside of the brackets must be multiplied by BOTH the a and the - 4 that are inside the brackets
5.
If 4(x - 5) = 10, what is the value of x?
5
7.5
10
15
4x - 20 = 10 and therefore 4x = 30
6.
3b - 16 + b = 1 is the same as which of the following?
4b = 1 - 16
4b = 1 / 16
4b = 1 + 16
4b = 1 x 16
When - 16 crosses the equals sign it changes to + 16
7.
If 5(x + 2) = -35, what is the value of x?
-5
-9
-10
10
5 x -7 = -35
8.
2x - 6 could be written as which of the following?
(x - 3)2
2(x - 3)
Either of the above
Neither of the above
Basically, 2x - 6 is the same as (x - 3) times 2
9.
Look at the following equation and choose the correct answer when the brackets have been expanded: 4(a + 3) = 5(b - 8)
4a - 12 = 5b - 40
4a - 12 = 5b + 40
4a + 12 = 5b - 40
4a + 12 = 5b + 40
We're multiplying the values in brackets by the numbers in front of them
10.
If 6a - 5 = 20, which of the following is incorrect?
6a = 25
a = 25 / 6
a = 416
a = 4.25
You will often find that you have a term such as 6a and you need to separate the 6 from the a in order to determine the value of a<|endoftext|>
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From time immemorial, humanity has always been curious about what lies beyond the skies. Thus, they created various astronomical devices and mechanisms that would allow them to observe and to somehow unravel the mysteries of celestial bodies.
One of the earliest structures made by ancient civilizations are the megaliths or the uniquely assembled massive stones that usually have circular formation and situated in wide open spaces. The building of these stone structures has continuously mystified scientists and archeologists of the modern ages. Studies led them to believe these large rock assemblages functioned as monuments for rituals, ceremonies, burials, and most especially as astronomical observatories and time-telling systems.
Amazingly, even without the use of telescopes or any advanced apparatus, people from the past were able to learn about the universe. They formed the boulders towards the alignment of celestial bodies – sun, moon, stars, and planets. It is believed this helped measure time and know the seasons for accomplishing specific activities such as planting and harvest. Indeed, construction of megaliths served a number of special purposes for the lives of early civilizations.
The oldest astronomical megaliths were discovered in Southeastern Turkey along the Taurus Mountains. This megalith is called the Göbekli Tepe and is believed to be built almost 12,000 years ago, 7,000 years before the construction of the famous Stonehenge in England. The stone pillars depict an orientation that reveals the cosmological beliefs of the inhabitants and builders of the monumental structures. Each stone column was positioned in a particular angle, which scientist concluded was to observe Deneb, the brightest star in Cygnus Constellation.
Another significant astronomical formation, believed to be built 11,000 years ago, is the Nabta Playa found at the Nubian Desert 100 km. West of Abu Simbel in the southernmost part of Egypt. It is located in the Tropic of Cancer, which is the region above the equator where the sun is directly overhead usually during the start of the summer season in the northern hemisphere. The complex arrangement of the stone slabs depict a symbolic geometrical architecture that aided the inhabitants during the beginning of rainy and dry seasons. It is believe they also represented the sun, water, and even death. Due to the shadow cast and the period when the megalith was constructed, caused scientists to believe that the Nabta culture inspired the construction of the pyramids.
There are several elaborately made astronomical megaliths all over the world that unravel the dynamic and impressive culture of ancient civilizations. Some of the oldest megaliths are in large parts of the Middle East and are assumed to be built during the early Mesopotamian period. These are called standing stones that function as places for religious rituals. European megaliths such as Stonehenge, Avebury, Ring of Brodgar and Beltany is known to be stone circles and believed to forecast lunar and solar activities. Asian megaliths, specifically in Northeast and Southeast Asia, were used primarily for funeral ceremonies called dolmens. Native American tribes in North and South America built megaliths and cliff palaces that aided their agricultural needs due to its location which was typically associated with sunrise.<|endoftext|>
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The vote-counting process in the 19th century was complex, unstandardized, and vulnerable to corruption. Counting practices varied from precinct to precinct, and a lot of alcohol was involved.
In the early part of the century, some Americans voted in public, using a method called viva voce (or “by voice”). This was a holdover from a Colonial practice, in which property owners (the only voters of the time) attended meetings and voted by a show of hands. In the 19th-century viva voce system, people went to local polling places and swore an oath that they were voting in good faith. Then, out loud and in front of anyone who cared to cluster around observing, the voter told the election judges his choices. The counting took place by hand; judges entered voter choices in poll books, keeping running totals of numbers of votes for each candidate. There were no paper ballots to tally.
In other locations, 19th-century voters used paper ballots issued by parties—a practice that became increasingly common as the century went on. Voters brought their own ballots to the polls, and although they could write their choices on pieces of paper, parties found that providing printed ballots with the names of their candidates was a convenience that nudged voters toward voting “straight ticket.” Parties printed their tickets on colored paper, and the ballots went into glass boxes, so that anyone observing a vote could clearly see which party a voter had chosen. The atmosphere at the polls was raucous, and party members lobbied for voters’ favor right up to the moment when they arrived at the ballot box.
As voters arrived at polling stations, election judges noted their names in poll books. At the end of the day, the tally of names was supposed to match up with the number of ballots collected. But in many locations, outnumbered judges, facing crowds of rowdy would-be voters, some of whom were illiterate and couldn’t spell their names, let the record-keeping slip. In some precincts in bigger cities, officials emptied ballot boxes at hourly intervals and checked numbers of ballots against the number of names on the books, which allowed for some small degree of quality control.
After the polls closed, the officials retired to a back room wherever the voting had taken place (often a saloon or tavern) to count ballots and cross-check them with the poll books. These judges, who were supposed to be impartial, were in many places appointed by police, who in turn were appointed by city officials—so local partisan politics inevitably crept into the counting process. In Southern towns during Reconstruction or Northern cities under the sway of political machines such as Tammany Hall, officials could “lose” ballots or write down incorrect numbers, and there were few checks against these practices.
The accuracy of the count may also have been affected by the officials’ tendency to extend Election Day partying into the night. Charles Albert Murdock, who lived in San Francisco in the 1860s and served as a poll judge, recalled in his memoir: “One served as an election officer at the risk of sanity if not of life. In the ‘fighting Seventh’ ward I once counted ballots for thirty-six consecutive hours, and as I remember conditions I was the only officer who finished sober.”
Until the end of the 19th century, the corruption inherent in both voting and ballot-counting was generally accepted as part of the game. But during the election of 1876, when Republican Rutherford B. Hayes faced Democrat Samuel J. Tilden, disputes over ballot counts in three Southern states—Florida, Louisiana, and South Carolina—delayed the results of the presidential race. Because Republican officials were in the majority on the panels that certified votes, they called for recounts and quickly declared that the states had actually gone for Hayes. Democrats contested the decision, going so far as to install alternate governors and state administrations rather than accept the legitimacy of the panels’ decisions. These recounts triggered a crisis on the federal level as Congress debated who held final authority to certify returns, and a president-elect wasn’t chosen for months. (Hayes won, but the vote-counting controversy was not without cost: The Republicans agreed to withdraw federal troops from Southern states, effectively ending Reconstruction.)
In response to the 1876 fiasco and to the influence of urban political machines, reformers called for the overhaul of the chaotic ballot system and succeeded in getting the so-called Australian ballot adopted in some forward-thinking states. In this system, the government printed a standardized ballot, which retains the secrecy of the voter’s choices while streamlining the counting process. There was still potential for corruption, however, since officials counting votes could willfully misinterpret voters’ checkmarks or stack tally teams with partisan allies willing to disqualify votes for the opposing candidate.
Mechanical lever voting machines, invented and adopted in the late 19th century, were supposed to circumvent the problems inherent in the hand-counted ballot, giving voters more secrecy while simplifying the counting process. The era of electronic vote-counting, which began in the middle of the 20th century, sped up election returns and regularized records, while bringing with it a new set of uncertainties—as anyone who followed the news in the fall of 2000 will recall.
Thanks to Jon Grinspan, curator of political history at the Smithsonian and author of The Virgin Vote: How Young Americans Made Democracy Social, Politics Personal, and Voting Popular in the Nineteenth Century, for his help.<|endoftext|>
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# What is the answer for question 16) ? http://postimg.org/image/6g52noj6f/
lemjay | Certified Educator
`log_3x+log_3(x+2)=1`
To explain this through the telephone, we can describe the steps as follows:
• Notice that the two logarithms in the equation have the same base. So, express it as one logarithm using the product property.
• So, at the left side of the equation write log_3 and multiply the two arguments x and (x+2). And the right side remains the same.
`log_3(x*(x+2)) = 1`
• Since `x*(x+2` ) is equal to `x^2+2x` , the argument of the logarithm becomes `x^2+2x` .
`log_3(x^2+2x) = 1`
• Since the logarithm of `x^2+2x` is equal to 1, then we apply the property that a logarithm is equal to 1 if its base and argument are the same `(log_b b=1)` .
• This means that the argument of `log_3` is equal to 3. So, set `x^2+2x` equal to 3. And this becomes our new equation.
`x^2+2x = 3`
• Now that we have a quadratic equation, to solve for x, set one side equal to zero. To do this, subtract both sides by 3.
`x^2+2x - 3 = 3-3`
`x^2+2x - 3 =0`
Then, factor left side.
`(x+3)(x - 1) = 0`
Next, set each factor equal to zero.
`x+ 3= 0` and `x-1=0`
So the equation breaks into two.
For the first equation, subtract both sides by 3.
`x+3=0`
`x+3-3=0-3`
`x=-3`
Note that in logarithm, we can not have a negative number as its argument. So, we do no consider -3 as a solution to our equation.
Next, solve for x in the second equation. To do so, add both sides by 1.
`x-1=0`
`x-1+1=0+1`
`x=1`
Hence, the solution to the given equation` log_3x + log_3(x+1)` is `x=1` .
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<meta http-equiv="refresh" content="1; url=/nojavascript/">
# Newton's Second Law
## The acceleration of an object equals the net force acting on the object divided by the object’s mass.
%
Progress
Practice Newton's Second Law
Progress
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Newton's Second Law
These boys are racing around the track at Newton’s Skate Park. The boy who can increase his speed the most will win the race. Tony, who is closest to the camera in this picture, is bigger and stronger than the other two boys, so he can apply greater force to his skates.
Q : Does this mean that Tony will win the race?
A : Not necessarily, because force isn’t the only factor that affects acceleration.
### Force, Mass, and Acceleration
Whenever an object speeds up, slows down, or changes direction, it accelerates. Acceleration occurs whenever an unbalanced force acts on an object. Two factors affect the acceleration of an object: the net force acting on the object and the object’s mass. Newton’s second law of motion describes how force and mass affect acceleration. The law states that the acceleration of an object equals the net force acting on the object divided by the object’s mass. This can be represented by the equation:
$\mathrm{Acceleration={\frac{Net \ force}{Mass}}}$
or $\mathrm{a={\frac{F}{m}}}$
Q : While Tony races along on his rollerblades, what net force is acting on the skates?
A : Tony exerts a backward force against the ground, as you can see in the Figure below , first with one skate and then with the other. This force pushes him forward. Although friction partly counters the forward motion of the skates, it is weaker than the force Tony exerts. Therefore, there is a net forward force on the skates.
### Direct and Inverse Relationships
Newton’s second law shows that there is a direct relationship between force and acceleration. The greater the force that is applied to an object of a given mass, the more the object will accelerate. For example, doubling the force on the object doubles its acceleration.
The relationship between mass and acceleration is different. It is an inverse relationship. In an inverse relationship, when one variable increases, the other variable decreases. The greater the mass of an object, the less it will accelerate when a given force is applied. For example, doubling the mass of an object results in only half as much acceleration for the same amount of force.
Q : Tony has greater mass than the other two boys he is racing (pictured in the opening image). How will this affect his acceleration around the track?
A : Tony’s greater mass will result in less acceleration for the same amount of force.
### Summary
• Newton’s second law of motion states that the acceleration of an object equals the net force acting on the object divided by the object’s mass.
• According to the second law, there is a direct relationship between force and acceleration and an inverse relationship between mass and acceleration.
### Explore More
At the following URL, use the simulator to experiment with force, mass, and acceleration. Click the "graph acceleration" and "graph applied force" buttons to display graphs for these values against time. Turn off friction in the toolbar to the right. Start with the textbook, set the slider next to the graph of applied force to its highest setting and then press go. Take a screenshot of your graphs to save them for later. Repeat this with several objects of different weights. Now compare the graphs of acceleration and applied force for each object with those of the other objects. Describe in words the relationship you see between mass and acceleration when the applied force is constant.
### Review
1. State Newton’s second law of motion.
2. How can Newton’s second law of motion be represented with an equation?
3. If the net force acting on an object doubles, how will the object’s acceleration be affected?
4. Tony has a mass of 50 kg, and his friend Sam has a mass of 45 kg. Assume that both friends push off on their rollerblades with the same force. Explain which boy will have greater acceleration.
### Vocabulary Language: English
Newton’s second law of motion
Newton’s second law of motion
Law stating that the acceleration of an object equals the net force acting on the object divided by the object’s mass.<|endoftext|>
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“This is a complicated navigational feat—it’s quite impressive for an animal that size,” said study co-author Eric Warrant, a biologist at the University of Lund in Sweden.
Moving in a straight line is crucial to dung beetles, which live in a rough-and-tumble world where competition for excrement is fierce. (Play “Dung Beetle Derby” on the National Geographic Kids website.)
Once the beetles sniff out a steaming pile, males painstakingly craft the dung into balls and roll them as far away from the chaotic mound as possible, often toting a female that they have also picked up. The pair bury the dung, which later becomes food for their babies.
But it’s not always that easy. Lurking about the dung pile are lots of dung beetles just waiting to snatch a freshly made ball. (Related: “Dung Beetles’ Favorite Poop Revealed.”)
That’s why ball-bearing beetles have to make a fast beeline away from the pile.
“If they roll back into the dung pile, it’s curtains,” Warrant said. If thieves near the pile steal their ball, the beetle has to start all over again, which is a big investment of energy.
Scientists already knew that dung beetles can move in straight lines away from dung piles by detecting a symmetrical pattern of polarized light that appears around the sun. We can’t see this pattern, but insects can thanks to special photoreceptors in their eyes.
But less well-known was how beetles use visual cues at night, such as the moon and its much weaker polarized light pattern. So Warrant and colleagues went to a game farm in South Africa to observe the nocturnal African dung beetle Scarabaeus satyrus. (Read another Weird & Wild post on why dung beetles dance.)
Attracting the beetles proved straightforward: The scientists collected buckets of dung, put them out, and waited for the beetles to fly in.
But their initial observations were puzzling. S. satyrus could still roll a ball in a straight line even on moonless nights, “which caused us a great deal of grief—we didn’t know how to explain this at all,” Warrant said.
Then, “it occurred to us that maybe they were using the stars—and it turned out they were.”
To test the star theory, the team set up a small, enclosed table on the game reserve, placed beetles in them, and observed how the insects reacted to different sky conditions. The team confirmed that even on clear, moonless nights, the beetles could still navigate their balls in a straight line.
To show that the beetles were focusing on the Milky Way, the team moved the table into the Johannesburg Planetarium, and found that the beetles could orient equally well under a full starlit sky as when only the Milky Way was present. (See Milky Way pictures.)
Lastly, to confirm the Milky Way results, the team put little cardboard hats on the study beetles’ heads, blocking their view of the sky. Those beetles just rolled around and around aimlessly, according to the study, published recently in the journal Current Biology.
Dung beetle researcher Sean D. Whipple, of the Entomology Department at the University of Nebraska-Lincoln, said by email that the “awesome results …. provide strong evidence for orientation by starlight in dung beetles.”
He added that this discovery reveals another potential negative impact of light pollution, a global phenomenon that blocks out stars.
“If artificial light—from cities, houses, roadways, etc.—drowns out the visibility of the night sky, it could have the potential to impact effective orientation and navigation of dung beetles in the same way as an overcast sky,” Whipple said.
Keep On Rollin’
Study co-author Warrant added that other dung beetles likely navigate via the Milky Way, although the galaxy is most prominent in the night sky in the Southern Hemisphere.
What’s more, it’s “probably a widespread skill that insects have—migrating moths might also be able to do it.”
As for the beetles themselves, they were “very easy to work with,” he added.
“You can do anything you want to them, and they just keep on rolling.”<|endoftext|>
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# Limits and Asymptotes
## Value the output of a function approaches as the input approaches a value; guideline representing a limit.
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Limits and Asymptotes
Suppose you stand exactly 4 feet from a wall, and begin moving toward the wall by halving the distance remaining with each step. How many steps would it take to actually get to the wall? How far would you walk in the process?
### Limits and Asymptotes
Consider the function f(x)=1x\begin{align*}f(x)=\frac{1}{x}\end{align*}. A graph of this function is shown below.
Notice that as the values of x get larger and larger, the graph gets closer and closer to the x-axis. In terms of the function values, we can say that as x gets larger and larger, f(x) gets closer and closer to 0. Formally, this kind of behavior of a function is called a limit. We say that as x approaches infinity, the limit of the function is 0. The line y = 0 is called the asymptote of the graph, it represents the value that f(x) will never quite reach. We can also say that f(x)=1x\begin{align*} f(x)=\frac{1}{x}\end{align*} is asymptotic to the line y = 0.
If we consider the behavior of the function as x approaches \begin{align*}-\infty\end{align*}, we see the same result: the limit of the function has x approaches \begin{align*}-\infty\end{align*} is also 0. Notice that this has the same asymptote: y = 0.
To be even more formal, we can write limits using a special notation. For the first limit, we write: limxf(x)=0.\begin{align*} \lim_{x \to \infty} f(x) = 0.\end{align*} For the second limit, we write limxf(x)=0.\begin{align*} \lim_{x \to -\infty} f(x) = 0.\end{align*} For any limit, here we will always write the x under the abbreviation “lim”, and then we will write the function under consideration. We can also write each of these limits with the specific function:
limx1x=0\begin{align*} \lim_{x \to \infty} \frac{1} {x} = 0\end{align*} and limx1x=0\begin{align*} \lim_{x \to -\infty} \frac{1} {x} = 0\end{align*}.
Because we are focused on end behavior, we are considering the limit of functions as x approaches ±\begin{align*}\pm \infty\end{align*}, and so the asymptotes we will find are horizontal lines. If we were examining other aspects of functions, we might find asymptotes that are vertical lines. For example, the function f(x)=1x\begin{align*} f(x)=\frac{1}{x}\end{align*} has a vertical asymptote at x = 0, or the y-axis. That is, the graph approaches the y-axis, as x values get closer and closer to 0.
### Examples
#### Example 1
Earlier, you were given a question about the distance involved in a strange walk towards a wall.
If you start 4ft from a wall, and halve the distance to it with each step, how many steps will it take, and how far will you walk, before you actually touch the wall?
Logically, we know that there is only a total distance of 4 feet between you and the wall, so no matter how you break it up, you cannot walk more than 4 feet. However, the actual distance you cover, and the number of steps it would take, cannot truly be defined since there could always be 1/2 of the remaining distance left. Technically speaking, you could continue the process forever without actually touching the wall! Of course, in practice, your ability to only move 1/2 of the remaining distance is limited by muscle control and measurement accuracy, so you would touch the wall before very many steps were actually taken.
Mathematically: limn(442n)=4\begin{align*}\lim_{n \to \infty} \left (4 - \frac{4} {2^n}\right) = 4\end{align*} , where n\begin{align*}n\end{align*} is the number of steps.
In other words: As the remaining distance gets closer and closer to 0, the total distance approaches 4.
#### Example 2
Write the limit described using limit notation.
The limit of some function f(x)\begin{align*}f(x)\end{align*} as x approaches infinity is 2.
We write the limit as follows:
limxf(x)=2\begin{align*}\lim_{x \to \infty} f(x) = 2\end{align*}
#### Example 3
Explain in words the meaning of the limit statement: limx(3+2x)=3\begin{align*}\lim_{x \to \infty} \left (3 + \frac{2} {x}\right) = 3\end{align*}
limx(3+2x)=3\begin{align*}\lim_{x \to \infty} \left (3 + \frac{2} {x}\right) = 3\end{align*} means: "As larger and larger numbers are substituted in for x\begin{align*}x\end{align*} in the function 3+2/x\begin{align*}3+2/x\end{align*}, the value comes closer and closer to 3\begin{align*}3\end{align*}.
This is due to the fact that the value added to 3\begin{align*}3\end{align*} gets smaller and smaller, down to effectively 0\begin{align*}0\end{align*} as 2\begin{align*}2\end{align*} is divided by larger and larger numbers.
#### Example 4
Determine the horizontal asymptote of the function g(x)=2x1x\begin{align*}g(x)=\frac{2x-1}{x}\end{align*} and express the asymptotic relationship using limit notation.
This function is asymptotic to the line y = 2.
The limit is written as limx2x1x=2\begin{align*}\lim_{x \to \neq \infty} \frac{2x - 1} {x} = 2\end{align*}.
We can determine the asymptote (and hence the limit) if we look at the graph. However, we can also analyze the equation to determine the limit. Consider the function g(x)=2x1x\begin{align*}g(x)=\frac{2x-1}{x}\end{align*}. As x approaches infinity, the x values are getting larger and larger. For sufficiently large values of x, the values of the expression 2x - 1 are very close to the values of the expression 2x, because subtracting one from a large number is fairly insignificant. Thus for sufficiently large values of x, 2x1x2xx2\begin{align*} \frac{2x-1}{x} \approx \frac{2x}{x} \approx 2 \end{align*}. As you can see from the accompanying table, which was created by a TI-83 graphing calculator, the function value gets closer to 2 as we look at larger and larger x values.
#### Example 5
Describe the following case and sketch a graph of the function with the given properties: limxf(x)=0\begin{align*}\lim_{x \to -\infty} f(x) = 0\end{align*}.
This reads: "The limit of f(x) as x approaches negative infinity is 0." In other words, as x gets massively negative, f(x) or y gets infinitely close to 0.
There are a number of possible graphs for this case; one example is offered below.
#### Example 6
Describe the following case and sketch a graph of the function with the given properties: limx4f(x)=3\begin{align*}\lim_{x \to 4} f(x) = 3\end{align*}.
This reads: "The limit of f(x) as x approaches 4 is 3." In other words, as x gets infinitely close to 4, f(x) or y gets infinitely close to 3. This can be a straight line, as y approaches 3 when 4 approaches 4 from either direction.
There are a number of possible graphs for this case; one example is offered below.
### Review
1. Define the terms horizontal asymptote and vertical asymptote.
2. Explain the difference between limx6f(x)=\begin{align*}\lim_{x \to -6} f(x) = \infty\end{align*} and limxf(x)=6\begin{align*}\lim_{x \to \infty} f(x) = -6\end{align*}.
3. Explain what limxf(x)=200\begin{align*}\lim_{x \to \infty} f(x) = 200\end{align*} means.
4. Explain what limx175f(x)=175\begin{align*}\lim_{x \to 175} f(x) = 175\end{align*} means.
Evaluate the following limits, if they exist. If a limit does not exist, explain why.
1. limt3t27tt8\begin{align*}\lim_{t \to \infty}\frac{3t^2 - 7t} {t-8}\end{align*}
2. \begin{align*}\lim_{t \to \infty} 3\end{align*}
3. \begin{align*}\lim_{t \to \infty}(t^2 - t^4)\end{align*}
4. \begin{align*}\lim_{x \to \infty} x +\sqrt {x^2 + 2x}\end{align*}
5. Find the horizontal and vertical asymptotes of the following function: \begin{align*}h(g) = \frac {5g^2 - 7g +9}{g^2 - 2g -3}\end{align*}
Given: \begin{align*} f(x) = \frac{x^2 - x - 6}{x^2 - 2x - 8}\end{align*} perform the following:
1. Find the horizontal and vertical asymptotes. Determine the behavior of \begin{align*} f \end{align*} near the vertical asymptotes.
2. Find the roots, y intercept and “holes” in the graph.
Determine \begin{align*}\lim_{t \to \infty} \frac{1}{t^n}\end{align*} if:
1. \begin{align*}n > 0\end{align*}
2. \begin{align*}n < 0\end{align*}
3. \begin{align*}n = 0\end{align*}
Let G & H be polynomials. Find \begin{align*}\lim_{x \to \infty} \frac{G(x)}{H(x)}\end{align*} if:
1. The degree of G is less than the degree of H
2. The degree of G is greater than the degree of H
3. The degree of G is the same as the degree of H
1. A pool contains 8000 L of water. An additive that contains 30g of salt per liter of water is added to the pool at a rate of 25 L per minute. a) Show that the concentration of salt after t minutes in grams per liter is: \begin{align*}C(t) = \frac{(t)30g\cdot25}{8000l + 25(t)l}\end{align*} b) What happens to the concentration as time increases to \begin{align*}\infty\end{align*}? Physically, why does this make sense?
To see the Review answers, open this PDF file and look for section 1.8.
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### Vocabulary Language: English
$\infty$
The symbol "$\infty$" means "infinity", and is an abstract concept describing a value greater than any countable number.
Asymptotes
An asymptote is a line on the graph of a function representing a value toward which the function may approach, but does not reach (with certain exceptions).
Asymptotic
A function is asymptotic to a given line if the given line is an asymptote of the function.
End behavior
End behavior is a description of the trend of a function as input values become very large or very small, represented as the 'ends' of a graphed function.
Horizontal Asymptote
A horizontal asymptote is a horizontal line that indicates where a function flattens out as the independent variable gets very large or very small. A function may touch or pass through a horizontal asymptote.
infinity
Infinity is an unbounded quantity that is greater than any countable number. The symbol for infinity is $\infty$.
limit
A limit is the value that the output of a function approaches as the input of the function approaches a given value.
Oblique Asymptote
An oblique asymptote is a diagonal line marking a specific range of values toward which the graph of a function may approach, but generally never reach. An oblique asymptote exists when the numerator of the function is exactly one degree greater than the denominator. An oblique asymptote may be found through long division.
Oblique Asymptotes
An oblique asymptote is a diagonal line marking a specific range of values toward which the graph of a function may approach, but generally never reach. An oblique asymptote exists when the numerator of the function is exactly one degree greater than the denominator. An oblique asymptote may be found through long division.
Piecewise Function
A piecewise function is a function that pieces together two or more parts of other functions to create a new function.
Slant Asymptote
A slant asymptote is a diagonal line marking a specific range of values toward which the graph of a function may approach, but will never reach. A slant asymptote exists when the numerator of the function is exactly one degree greater than the denominator. A slant asymptote may be found through long division.
Vertical Asymptote
A vertical asymptote is a vertical line marking a specific value toward which the graph of a function may approach, but will never reach.<|endoftext|>
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Home » Revision Notes for CBSE Class 6 to 12 » CBSE Class 8 Maths Revision Notes Chapter 15 – Introduction to Graphs
# CBSE Class 8 Maths Revision Notes Chapter 15 – Introduction to Graphs
## Revision Notes for CBSE Class 8 Maths Chapter 15 – Free PDF Download
Free PDF download of Class 8 Maths Chapter 15 – Introduction to Graphs Revision Notes & Short Key-notes prepared by expert Maths teachers from latest edition of CBSE(NCERT) books. All Chapter 15 – Introduction to Graphs Revision Notes to help you to revise complete Syllabus and Score More marks.
Maths NCERT Solutions for Class 8
Chapter Name Introduction to Graphs Chapter Chapter 15 Class Class 8 Subject Maths Revision Notes Board CBSE TEXTBOOK CBSE NCERT Category Revision Notes
## Quick Revision Notes
• Graphical presentation of data is easier to understand.
(i) A bar graph is used to show comparison among categories.
(ii) A pie graph is used to compare parts of a whole.
(iii) A Histogram is a bar graph that shows data in intervals.
• A line graph displays data that changes continuously over periods of time.
• A line graph which is a whole unbroken line is called a linear graph.
• For fixing a point on the graph sheet we need, x-coordinate and y-coordinate.
• The relation between dependent variable and independent variable is shown through a graph.
• A Bar Graph: A pictorial representation of numerical data in the form of bars (rectangles) of uniform width with equal spacing. The length (or height) of each bar represents the given number.
• A Pie Graph: A pie graph is used to compare parts of a whole. The various observations or components are represented by the sectors of the circle.
• A Histogram: Histogram is a type of bar diagram, where the class intervals are shown on the horizontal axis and the heights of the bars (rectangles) show the frequency of the class interval, but there is no gap between the bars as there is no gap between the class intervals.
• Linear Graph: A line graph in which all the line segments form a part of a single line.
• Coordinates: A point in Cartesian plane is represented by an ordered pair of numbers.
• Ordered Pair: A pair of numbers written in specified order.<|endoftext|>
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